Second joint Conference of the Belgian, Royal Spanish and ... · Second joint Conference of the...

Second joint Conference ofthe Belgian, Royal Spanish

and LuxembourgMathematical Societies

Logrono, 6 - 8 June, 2016

Book of Abstracts

and List of Participants

Contents

Plenary Talks 11

A glimpse of the Langlands programmeSara Arias de Reyna . . . . . . . . . . . . . . . . . . . . . . . . . 11

Math between two beautiful bracketsMarıa Jesus Carro . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Real, p-adic, and motivic (oscillatory) integrals and applicationsRaf Cluckers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Props of ribbon graphs, involutive Lie bialgebras and moduli spaces ofcurvesSergei Merkulov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

From elementary number theory to string theory and back againJohannes Nicaise . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Forests, Trees, Words, LettersJesus Marıa Sanz-Serna . . . . . . . . . . . . . . . . . . . . . . . 16

Brownian motion, Ricci curvature and entropyAnton Thalmaier . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Session 1: Functional Analysis 18

Maximally divergent Fourier series in the disc algebraLuis Bernal Gonzalez . . . . . . . . . . . . . . . . . . . . . . . . 18

Weak Banach-Saks and Radon-Nikodym properties in function spacesGuillermo P. Curbera . . . . . . . . . . . . . . . . . . . . . . . . 19

On the non-triviality of certain spaces of analytic functions. Ultrahy-perfunctions and hyperfunctions of fast growthAndreas Debrouwere . . . . . . . . . . . . . . . . . . . . . . . . . 20

Multifractal analysis of the divergence of wavelet seriesCeline Esser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Banach spaces admitting many complemented subspacesManuel Gonzalez . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

On upper frequent hypercyclicityKarl Grosse-Erdmann . . . . . . . . . . . . . . . . . . . . . . . . 23

Periodic points at the service of hypercyclicityQuentin Menet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Integral operators mapping into H∞

Jose Angel Pelaez . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Additivity of 2-local and weak-2-local maps on C∗-algebrasAntonio M. Peralta . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3

4 CONTENTS

Set-Valued Chaos in Linear DynamicsAlfred Peris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

My friend Manuel ValdiviaJean Schmets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Isomorphic copies of `1 for m-homogeneous non-analytic Bohnenblust-Hille polynomialsJuan B. Seoane-Sepulveda . . . . . . . . . . . . . . . . . . . . . . 29

Session 2: Model Theory and Applications 30

On dependently embedded setsEnrique Casanovas . . . . . . . . . . . . . . . . . . . . . . . . . . 30

P -minimality and p-adic integrationPablo Cubides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Expansion of valued fields by multiplicative subgroupNathanael Mariaule . . . . . . . . . . . . . . . . . . . . . . . . . 32

A simple remarkAmador Martın-Pizarro . . . . . . . . . . . . . . . . . . . . . . . 33

Superrosy division ringsDaniel Palacın . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Oscillatory integrals and o-minimalityT. Servi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Geometric constructions as a transfer tool in Tropical GeometryLuis F. Tabera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

On generalized Goodstein sequencesA. Weiermann . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Session 3: Algebra and Number Theory 38

Artin and Hilbert type theorems for Lie algebrasAna Agore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Gradings on exceptional simple Jordan systems and structurable alge-brasDiego Aranda-Orna . . . . . . . . . . . . . . . . . . . . . . . . . 39

On bialgebroids and warpingsMitchell Buckley . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

On the Prime Graph Question for symmetric groupsMauricio Caicedo . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Units in integral group rings via fundamental domains and hyperbolicgeometryAnn Kiefer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Weak wreath products and weak quantum duplicatesEsperanza Lopez Centella . . . . . . . . . . . . . . . . . . . . . . 43

Darmon points on modular abelian varieties over totally real fieldsSantiago Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Homological torsion originating from finite subgroupsAlexander D. Rahm . . . . . . . . . . . . . . . . . . . . . . . . . 45

Nonmatrix varieties for some classes of non associative algebrasIvan Shestakov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

The self-normalizing case of the McKay conjectureCarolina Vallejo Rodrıguez . . . . . . . . . . . . . . . . . . . . . 47

CONTENTS 5

On the Distribution of Frobenius of Weight 2 Eigenforms with QuadraticCoefficient FieldJasper Van Hirtum . . . . . . . . . . . . . . . . . . . . . . . . . 48

Session 4: Partial Differential Equations 49

Standing wave solutions for a nonlinear Scrodinger equation with mixeddispersionJean-Baptiste Casteras . . . . . . . . . . . . . . . . . . . . . . . . 49

Nonlinear elliptic singular systems with quadratic gradient lower ordertermsJose Carmona . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Liouville type equations with sign-change dataRafael Lopez-Soriano . . . . . . . . . . . . . . . . . . . . . . . . . 51

On the eigenvalues of Aharonov–Bohm operators with varying polesManon Nys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

On the extended Allen–Cahn equationAlberto Saldana . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Some problems arising in population dynamics with non-linear diffu-sionAntonio Suarez . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Session 5: Algebraic Geometry and Singularities 55

Constancy regions of mixed multiplier ideals in rational surface singu-laritiesMarıa Alberich Carraminana . . . . . . . . . . . . . . . . . . . . 55

Geometric invariants encoded in the Newton polygonWouter Castryck . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Splicing and zeta functionsThomas Cauwbergs . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Topology of spaces of valuations and geometry of singularitiesAna Belen de Felipe . . . . . . . . . . . . . . . . . . . . . . . . . 58

The arc space of the GrassmannianRoi Docampo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Javier Fernandez de Bobadilla . . . . . . . . . . . . . . . . . . . 60

Nearby cycles and Alexander modules of hypersurface complementsYongqiang Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

On the 2-Jordan blocks for the eigenvalue λ = 1 of isolated singularitiesJorge Martın Morales . . . . . . . . . . . . . . . . . . . . . . . . 61

On the estimation of numerical invariants of a graded module in termsof its Hilbert seriesJulio Jose Moyano Fernandez . . . . . . . . . . . . . . . . . . . . 62

Positivity of divisors on blown-up projective spacesElisa Postinghel . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Duality on value semigroupsMathias Schulze . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A family of symplectic-complex Calabi-Yau manifolds that are nonKahlerBotong Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 CONTENTS

Session 6: Dynamical Systems and ODE 66

Coexistence of hyperchaos and chaos: a Computer-assisted proofRoberto Barrio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A journey in the zoo of Turing patternsTimoteo Carletti . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Space Debris in the Geostationary region as a Dynamical SystemDaniel Casanova . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Slow-fast Bogdanov-Takens bifurcations in an applicationPeter De Maesschalck . . . . . . . . . . . . . . . . . . . . . . . . 70

The role of chaos in the formation of binary objects in the Kuiper-beltDavid Farrelly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Summability of canard-heteroclinic saddle connectionsKarel Kenens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Exterior discrete semi-flows and applicationsMiguel Maranon . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Dissipativity in nonautonomous linear-quadratic control processesCarmen Nunez . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

The parabolic RTBP. Interchange of mass after a close encounter be-tween galaxiesMerce Olle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Persistence of non-autonomous Nicholson’s systemsAna M. Sanz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Expanding Baker Maps: A class of piecewise linear 2-D mapsEnrique Vigil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Normal forms near a symmetric planar saddle connectionJeroen Wynen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Session 7: Probability and Statistics 79

Spatio-temporal P -splines models in Bayesian disease mappingAritz Ardin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Recent developments in Stein’s methodChristian Dobler . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Nonparametric conditional copula estimation and applicationsIrene Gijbels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Targeted penalized regression for cost effective causal effect estimationEls Goetghebeur . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A continuous updating weighted least squares estimator of tail depen-dence in high dimensionsAnna Kiriliouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

SAS distributionsArthur Pewsey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Non-Universality of Nodal Length Distribution for Arithmetic RandomWavesMaurizia Rossi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Circular Isotonic Regression with Applications to cell-cycle BiologyCristina Rueda . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Integrated likelihood based inference for nonlinear panel data modelswith unobserved effectsMartin Schumann . . . . . . . . . . . . . . . . . . . . . . . . . . 87

CONTENTS 7

An applied approach to robust statistical analysis of the location ofinterval-valued dataBeatriz Sinova . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Stein’s method and the influence of the prior in Bayesian statisticsYvik Swan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Heat kernel formulae and the Brownian bridge to a submanifoldJames Thompson . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Session 8: Orthogonal Polynomials and Special Functions 91

Weighted inequalities for the Riesz potential on the sphereAlberto Arenas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Critical points of polynomialsManuel Bello . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Darboux for CMV and Christoffel transformationsMarıa Jose Cantero . . . . . . . . . . . . . . . . . . . . . . . . . 93

Positive quadrature formulas on the unit circle with prescribed nodes.A new approachRuyman Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A higher rank generalization of the Bannai-Ito algebraHendrik De Bie . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Orthogonal polynomials, partition functions and asymptoticsAlfredo Deano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Asymptotics of Sobolev Orthogonal polynomials on the unit ballAntonia Delgado . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

An analogue of Krall orthogonal polynomials on the simplexLidia Fernandez . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Christoffel transformations for matrix orthogonal polynomials in thereal lineJuan Carlos Garcıa Ardila . . . . . . . . . . . . . . . . . . . . . . 99

A Hardy inequality for ultraspherical expansions with an applicationto the sphereEdgar Labarga . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Dynamics and interpretation of some integrable systems via matrixorthogonal polynomialsAna Mendes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Dries Stivigny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Session 9: Combinatorial and Computational Geometry 104

Switched symplectic graphs and their 2-ranksAida Abiad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Symmetric generalised polygonsJohn Bamberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

The finiteness threshold width of lattice polytopesMonica Blanco . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Cameron-Liebler sets of generators in polar spacesMaarten De Boeck . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Infinite families of non linear MRD codesNicola Durante . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8 CONTENTS

Extensions of Minkowski’s theorem on successive minimaM Angeles Hernandez Cifre . . . . . . . . . . . . . . . . . . . . . 109

Production matrices for geometric graphsClemens Huemer . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

The Manickam-Miklos-Singhi Conjecture in Partial Linear SpacesFerdinand Ihringer . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Alberto Marquez . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Domination via (min,+) algebraMari Luz Puertas . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Rectilinear Convex Hull of a Set of Points in 2DCarlos Seara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Session 10: Geometric Analysis, Differential Geometry and Quan-tization 114

Pierre Bielavsky . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Feynman Integrals, Associated Arrangements and Their MotivesOzgur Ceyhan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Isoparametric hypersurfaces in complex hyperbolic spacesJose Carlos Dıaz . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Hofer’s geometry of the space of curvesMichael Khanevsky . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Moment maps and closed Fedosov’s star productsLaurent La Fuente Gravy . . . . . . . . . . . . . . . . . . . . . . 117

Lagrangian mean curvature fow of Hopf toriAna Lerma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

The space of oriented spheres as a bridge between H3 and R3

Antonio Martınez . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Graded geometry in physics and mechanicsVladimir Salnikov . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Pre-symplectic structures and related deformation problemsFlorian Schaetz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Kerr-Schild vector fieldsJose M. Senovilla . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Constant mean curvature surfaces in Riemannian product spacesFrancisco Torralbo . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Posters 124

Local Nilpotency of the McCrimmon Radical of a Jordan SystemJose Anquela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Some results about diametral dimensionsFranoise Bastin, Loıc Demeulenaere, Leonhard Frerick, JochenWengenroth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

On differential Galois groups of strongly normal extensionsQuentin Brouette, Franoise Point . . . . . . . . . . . . . . . . . . 126

Preliminaries to the study of Radon-type transforms in a symplecticframeworkMichel Cahen, Thibaut Grouy, Simone Gutt . . . . . . . . . . . . 127

CONTENTS 9

Marginally trapped submanifolds in generalized Robertson-Walker space-timesV. L. Canovas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

A first approach to security analysis on networks with ancilliary pointsM. Carriegos-Vieira, M.T. Trobajo . . . . . . . . . . . . . . . . . 129

Some applications of the non-abelian tensor product of Hom-LeibnizalgebrasJ. M. Casas, E. Khmaladze, N. Pacheco Rego . . . . . . . . . . . 130

Plane curves whose curvature depends on distance from a pointIldefonso Castro, Ildefonso Castro-Infantes, Jesus Castro-Infantes 131

Umbilicity properties of spacelike codimension two submanifoldsNastassja Cipriani, JosMarıa Martın Senovilla, Joeri Van derVeken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

The Kurosh Problem for Jordan Nil Systems over Arbitrary Rings ofScalarsTeresa Cortes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

On the generalization of analytic local loopsDaniel de la Concepcion . . . . . . . . . . . . . . . . . . . . . . . 134

Weighted Hardy inequalities, real interpolation methods and vectormeasuresR. del Campo, A. Fernandez, A. Manzano, F. Mayoral, F. Naranjo135

From elliptic functions to modular formsMarıa de las Mercedes Ganim . . . . . . . . . . . . . . . . . . . . 136

Some ideas to improve the domain of starting points for Newton?smethodJ.A. Ezquerro, M.A. Hernandez . . . . . . . . . . . . . . . . . . 137

Some variants of continuous Newton’s methodJ.M. Gutierrez, M.A. Hernandez . . . . . . . . . . . . . . . . . . 138

Dynamical study of the Secant methodJ.M. Gutierrez, A.A. Magrenan . . . . . . . . . . . . . . . . . . . 139

A capture model of Jupiter’s irregular moons based on a restricted2 + 2 body problemW. Kanaan, V. Lanchares . . . . . . . . . . . . . . . . . . . . . . 140

From quantum moment maps to deformation quantizations on boundedsymmetric domainsStephane Korvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Superstable expansions of (Z,+, 0)Quentin Lambotte . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Finite metric planes and related objectsJesse Lansdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A boundedness Nikodym property in algebras of Jordan measurablesetsSalvador Lopez Alfonso . . . . . . . . . . . . . . . . . . . . . . . 145

Topological groups and spaces C(X) with ordered basesM. Lopez-Pellicer, J.C. Ferrando, J. Kakol . . . . . . . . . . . . 146

Asymptotic behavior of varying discrete Sobolev orthogonal polyno-mialsJ.F. Manas-Manas, F. Marcellan, J.J. Moreno-Balcazar . . . . . 147

Area estimates for constant mean curvature surfaces in E(κ, τ)-spacesJose M. Manzano, Barbara Nelli . . . . . . . . . . . . . . . . . . 148

10 CONTENTS

Orthogonality for generalized Gegenbauer weight functions on the ballwith an extra term on the sphereClotilde Martınez, Miguel A. Pinar . . . . . . . . . . . . . . . . . 149

Some sums of powers of Catalan triangle numbersPedro J. Miana, Hideyuki Ohtsuka, Natalia Romero . . . . . . . 150

On a Moser-Kurchatov type method for nonlinear equationsMarıa Jesus Rubio . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Remarks on the set of norm-attaining functionals and differentiabilityV. Montesinos, A.J. Guirao, V. Zizler . . . . . . . . . . . . . . . 152

Convergent expansions of special functions in terms of elementary func-tionsP.J. Pagola, B. Bujanda, J.L. Lopez . . . . . . . . . . . . . . . 153

On Topological Exponential Differential FieldsNathalie Regnault . . . . . . . . . . . . . . . . . . . . . . . . . . 154

On linear refinements of geometric inequalitiesJ. Yepes Nicolas . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

II Joint Conference of the Belgian, Royal Spanish and Luxembourg Mathematical Societies

Logrono, June 6–8, 2016

A glimpse of the Langlands programme

Sara Arias de Reyna1

The Langlands programme consists of a series of conjectures relating arithmeticobjects (number fields) to analytic objects (automorphic forms), and it constitutes acentral theme in modern number theory.

In this talk we will approach the Langlands programme through the followingquestion: Given a polynomial f(x) with integer coefficients, for which prime numbersp does it split into different linear factors modulo p?

We will review advances on this question, from Gauss reciprocity law to class fieldtheory and its non-abelian generalisations (some already proven, some conjectured),stopping by the proof of Fermat’s Last Theorem.

1Faculte des Sciences, de la Technologie et de la CommunicationUniversite du Luxembourgsara [email protected]

11



Math between two beautiful brackets

Marıa Jesus Carro1

I shall not say, in advance, what the beautiful brackets will be but those resultsreferring to maths will be dealing with a powerful technique in Harmonic Analysisdealing with boundedness of operators in weighted Lebesgue space. Brackets arerelated to open intervals and on many occasions to reach the extreme is one of themost intriguing goal.

1Departamento de Matematica Aplicada y Anlisis Facultad de MatematicasUniversidad de [email protected]

12



Real, p-adic, and motivic (oscillatory) integrals andapplications

Raf Cluckers1

In the real, p-adic and motivic settings, we will present recent results on oscillatoryintegrals, from a rather modern viewpoint. In the reals, they are related to subanalyticfunctions and their Fourier transforms, and questions related to o-minimality arize.In the p-adic and motivic case, there are furthermore transfer principles (to switchbetween fields of different characteristics) which lead to applications in representationtheory and the Langlands program. This is joint work with Comte, Gordon, Hales,Halupczok, Loeser, Miller, Rolin, and Servi (in various combinations).

1Universit Lille 1 Sciences et TechnologiesLaboratoire [email protected]

13



Props of ribbon graphs, involutive Lie bialgebrasand moduli spaces of curves

Sergei Merkulov1

We discuss a new and surprisingly strong link between two previously unrelatedtheories: the theory of moduli spaces of curves (which, according to Penner, is con-trolled by the ribbon graph complex) and the homotopy theory of operads (controlledby ordinary graph complexes with no ribbon structure, introduced first by Kontse-vich). The talk is based on a joint work with Thomas Willwacher.

1Faculte des Sciences, de la Technologie et de la CommunicationUniversite du [email protected]

14



From elementary number theory to string theoryand back again

Johannes Nicaise1

I will describe some surprising interactions between number theory, algebraic ge-ometry and mirror symmetry that have appeared in my recent work with MirceaMustata and Chenyang Xu and that have led to a solution of Veys? 1999 conjectureon poles of maximal order of Igusa zeta functions. The talk will be aimed at a generalaudience and will emphasize some key ideas from each of the fields involved ratherthan the technical aspects of the proof.

1Imperial College [email protected]

15



Forests, Trees, Words, Letters

Jesus Marıa Sanz-Serna1

The construction and analysis of some numerical integrators is a surprisingly com-plicated task. Several decades were necessary before numerical analystis came up withthe necessary tools from algebra and combinatorics. Later these tools have found usesin other fields like renormalization in quantum field theory, noncommutative geom-etry, regularity structures, etc. This is a talk aimed at nonspecialists, where I shallpresent a historical survey, a summary of the main tools and a sketch of recent devel-opments.

1Departamento de MatematicasUniversidad Carlos [email protected]

16



Brownian motion, Ricci curvature and entropy

Anton Thalmaier1

We discuss notions of stochastic differential geometry in the framework of staticmanifolds and manifolds evolving along a geometric flow. In this context we deal withBrownian motions, basic functional inequalities, and entropy formulas for positivesolutions of heat equations.

1University of [email protected]

17



Maximally divergent Fourier series in the discalgebra

Luis Bernal-Gonzalez

From Carleson’s theorem it is derived as a special case that a continuous functionT → C on the unit circle T of the complex plane C is the a.e. sum of its Fourierseries. Hence divergence can only be expected at small subsets of T. Existence of pe-riodic continuous functions having divergent Fourier series on prescribed null-measuresubsets is well known. In this talk –whose results form a joint work with J. Mullerand A. Jung– it is shown that there are large vector spaces of functions in the discalgebra A(D) such that every nonzero member satisfies that the restriction to T ofits Fourier series is maximally divergent, in the sense that, for many small subsets Eof T, its partial sums approximate any prescribed function on E. This completes orimproves a number of findings by several authors.

Department of Analisis Matematico, University of Sevilla, Apartado 1160,Avenida Reina Mercedes, 41080 Sevilla, [email protected]

18



Weak Banach-Saks and Radon-Nikodym propertiesin function spaces

Guillermo P. Curbera

A classical result of Hardy motivated the study of the Cesaro operator

C : f 7→ C(f)(x) :=1

x

∫ x

0

f(t) dt,

in the Lp spaces, leading to the spaces Cesp := f : C(|f |) ∈ Lp, which have beenextensively studied. We focus our attention on the spaces obtained by consideringoperators T other than the Cesaro operator and function spaces X other than Lp,resulting in the spaces

[T,X] :=f : T (|f |) ∈ X

.

We consider the weak Banach-Saks and the Radon-Nikodym properties for thespaces [T,X].

The results presented in this talk are joint work with Werner J. Ricker from theKatholische Universitat Eichstatt-Ingolstadt (Germany).

Facultad de Matematicas, Universidad de Sevilla, Aptdo. 1160, 41080 [email protected]

19



On the non-triviality of certain spaces of analyticfunctions. Ultrahyperfunctions and hyperfunctions

of fast growth.

Andreas Debrouwere

The test function spaces for Fourier ultrahyperfunctions and Fourier hyperfunc-tions (in one dimension) consist of functions ϕ which are analytic on the horizontalstrip | Im z| < k and satisfy

sup| Im z|<k

|ϕ(z)|ek|z| < ∞,

for each k > 0 and some k > 0, respectively. In this talk we are interested in thefollowing generalization of these spaces: let M be a non-decreasing positive functiondefined on the positive half-axis satisfying certain natural conditions – e.g. the as-sociated function of a weight sequence Mp – and consider the spaces consisting offunctions ϕ which are analytic on the strip | Im z| < k and satisfy

sup| Im z|<k

|ϕ(z)|eM(k|z|) < ∞,

for each k > 0 and some k > 0, respectively. Our results are twofold. Firstly, wepresent an analytic representation theory for the duals of these spaces and expressthem as cohomological quotients of spaces of analytic functions. Secondly, by usingthe aforementioned results, we characterize the non-triviality of these test functionspaces in terms of the growth of the weight function M . In particular, we show

that the Gelfand-Shilov spaces of Beurling type S(p!)(Mp)

and Roumieu type Sp!Mp are

non-trivial if and only if

supp≥2

(log p)p

hpMp< ∞,

for all h > 0 and some h > 0, respectively.

Department of Mathematics, Ghent University, 9000 Gent, Belgium, [email protected]

20



Multifractal analysis of the divergence of waveletseries

Celine Esser1, Stephane Jaffard2

In this talk, we study pointwise divergence properties of wavelet expansions offunctions in a given Besov space. We obtain deterministic upper bounds for theHausdorff dimensions of the sets of points where a given rate of divergence is observed,and we show that these bounds are generically (in the sense of Baire’s categories)optimal. This gives a complement to the works done by F. Bayart and Y. Heurteaux(cf. [2]) in the case of Fourier series and by J.M. Aubry (cf. [1]).

References

[1] J.M. Aubry, On the rate of pointwise divergence of Fourier and wavelet series inLp, J. Approx. Theory 538 (2006), 97–111.

[2] F. Bayart and Y. Heurteaux, Multifractal analysis of the divergence of Fourierseries, Ann. Sci. Ec. Norm. Super. 45 (2012), 927–946.

1Universite Lille 1Laboratoire Paul Painleve, Bat. M359655 Villeneuve d’Ascq, [email protected]

2Universite Paris-Est – Creteil Val-de-Marne, Bat. P2Avenue du General de Gaulle, 6194010 Creteil, [email protected]

21



Banach spaces admittingmany complemented subspaces

Manuel Gonzalez

A Banach space X is called subprojective if every (closed) infinite-dimensionalsubspace of X contains an infinite-dimensional subspace complemented in X, and Xis called superprojective if every infinite-codimensional subspace of X is contained inan infinite-codimensional subspace complemented in X. These two classes of spaceswere introduced by Whitley [6] to find conditions for the conjugate of an operatorto be strictly singular or strictly cosingular, and they were applied in [2] to obtainsolutions to the perturbation classes problem for semi-Fredholm operators.

Banach spaces in which every subspace is complemented are isomorphic to Hilbertspaces [4], but a significant number of spaces are subprojective or superprojective. Forexample, Lp(0, 1) is subprojective if and only if 2 ≤ p <∞, and it is superprojectiveif and only if 1 < p ≤ 2.

In this talk we describe some recent results on subprojective and superprojectivespaces obtained in [5], [3] and [1].

References

[1] E. M. Galego, M. Gonzalez and J. Pello. On subprojectivity and super-projectivity of Banach spaces. Preprint 2016.

[2] M. Gonzalez, A. Martınez-Abejon and M. Salas-Brown. Perturbationclasses for semi-Fredholm operators on subprojective and superprojective Banachspaces. Ann. Acad. Sci. Fennicae Math. 36 (2011), 481–491.

[3] M. Gonzalez and J. Pello. Superprojective Banach spaces. J. Math. Anal.Appl. 437 (2016), 1140–1151.

[4] J. Lindenstrauss and L. Tzafriri. On the complemented subspaces problem.Israel J. Math. 9 (1971), 263–269.

[5] T. Oikhberg and E. Spinu. Subprojective Banach spaces. J. Math. Anal. Appl.424 (2015), 613–635.

[6] R. J. Whitley. Strictly singular operators and their conjugates. Trans. Amer.Math. Soc. 113 (1964), 252–261.

Departamento de Matematicas, Universidad de Cantabria, Avenida de losCastros s/n, E-39071 Santander, [email protected]

22



On upper frequent hypercyclicity

Karl Grosse-Erdmann1,

Hypercyclic and frequently hypercyclic operators play a central role in linear dy-namics. However, while the set of hypercyclic vectors is always residual (unless itis empty), the set of frequently hypercyclic vectors is always meagre. Bayart andRuzsa [1] have recently shown that the set of upper frequently hypercyclic vectorsis, again, always residual (or empty). We investigate in greater detail why this isso. Our results have consequences, among other things, for the notion of reiterativehypercyclicity that was recently introduced by Bes, Menet, Peris and Puig [2].

This is joint work with Antonio Bonilla [3].

References

[1] F. Bayart and I. Z. Ruzsa, Difference sets and frequently hypercyclic weightedshifts, Ergodic Theory Dynam. Systems 35 (2015), 691–709.

[2] J. Bes, Q. Menet, A. Peris, and Y. Puig, Recurrence properties of hyper-cyclic operators, Math. Ann., to appear.

[3] A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity andrelated notions, arXiv:1601.07276.

1Departement de Mathematique, Universite de Mons, 20 Place du Parc,7000 Mons, [email protected]

23



Periodic points at the service of hypercyclicity

Menet Quentin1

If a vector possesses a dense orbit, we say that the vector is hypercyclic. On theother hand, a periodic point for an operator T is a vector x such that T dx = x forsome d ≥ 1. It can seem a little bit surprising but the existence of periodic pointscan be useful for obtaining hypercyclic vectors. The goal of this talk will consist inshowing how the existence of a dense set of periodic points satisfying some propertiescan be used for constructing hypercyclic vectors, U -frequently hypercyclic vectors andfrequently hypercyclic vectors. These criteria can for instance be used for finding newcounterexamples in linear dynamics.

1Laboratoire de Mathematiques de Lens (LML), Universite d’Artois,Rue Jean Souvraz SP 18, F-62300 Lens, [email protected]

24



Integral operators mapping into H∞

Jose Angel Pelaez1,

We address the problem of studying the boundedness, compactness and weak com-pactness of the integral operators Tg(f)(z) =

∫ z

0f(ζ)g′(ζ) dζ acting from a Banach

space X into H∞. We obtain a collection of general results which are appropriatelyapplied and mixed with specific techniques in order to solve the posed questions toparticular choices of X.

Joint work with Manuel D. Contreras, Christian Pommerenke and Jouni Rattya.

1Departamento de Analisis MatematicoFacultad de Ciencias29071, Malaga, [email protected]

25



Additivity of 2-local and weak-2-local maps onC∗-algebras

Antonio M. Peralta1,

Additivity of certain maps defined on C∗-algebras seems to be closely relatedto other algebraic properties determined by the Algebraic-Analytic structure of C∗-algebras. That is the case of 2-local and weak-2-local maps defined by 2-local actionsof derivations and ∗-homomorphisms on C∗-algebras.

We shall present the latest advances on 2-local and weak-2-local maps on C∗-algebras. Let S be a subset of the space L(X,Y ) of all linear maps between Banachspaces X and Y . We recall that a (non-necessarily linear nor continuous) mapping∆ : X → Y is a 2-local S map (respectively, a weak-2-local S map) if for each x, y ∈ X(respectively, if for each x, y ∈ X and φ ∈ Y ∗), there exists Tx,y ∈ S, depending on xand y (respectively, Tx,y,φ ∈ S, depending on x, y and φ), satisfying

∆(x) = Tx,y(x), and φ∆(y) = Tx,y(y)

(respectively,φ∆(x) = φTx,y,φ(x), and φ∆(y) = φTx,y,φ(y)).

We shall mainly focus on the case in which X and Y are C∗-algebras and S is theset of all derivations on A or the set of all ∗-homomorphisms from A to B, or simplythe set of all symmetric linear maps between A and B. We shall derive generalizationsof well known, and significant, results due to Kowalski and S lodkowski [2], Semrl [3],and Ayupov and Kudaybergenov [1].

References

[1] Sh. Ayupov and K.K. Kudaybergenov, 2-local derivations on von Neumannalgebras, Positivity 19 (2015), No. 3, 445-455.

[2] S. Kowalski and Z. S lodkowski, A characterization of multiplicative linearfunctionals in Banach algebras, Studia Math. 67 (1980), 215-223.

[3] P. Semrl, Local automorphisms and derivations on B(H), Proc. Amer.Math. Soc. 125 (1997), 2677-2680.

1Departamento de Analisis Matematico, Universidad de Granada, Facultadde Ciencias 18071, Granada, [email protected]

Author partially supported by the Spanish Ministry of Economy and Compet-itiveness and European Regional Development Fund project no. MTM2014-58984-P and Junta de Andalucıa grant FQM375.

26



Set-Valued Chaos in Linear Dynamics

N. C. Bernardes Jr.1, A. Peris2, F. Rodenas2

We will present several notions of chaos for hyperspace dynamics associated tolinear operators. More precisely, we consider a linear operator T : X → X on atopological vector space X, and the natural hyperspace extensions T and T of T tothe spaces K(X) of compact subsets of X and C(X) of convex compact subsets ofX, respectively, endowed with the Vietoris topology. We show that the topologicaltransitivity of T or T is equivalent to the weak mixing property of T , extending resultsin [1] and [4]. Analogous results are obtained for topological ergodicity and mixing.When X is a Frechet space, then Devaney chaos is equivalent for the maps T , T andT . Finally, under very general conditions, we obtain the corresponding equivalencesfor Li-Yorke chaos.

References

[1] J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals 25 (2005),no. 3, 681–685.

[2] W. Bauer and K. Sigmund, Topological dynamics of transformations induced onthe space of probability measures, Monatsh. Math. 79 (1975), 81–92.

[3] G. Herzog and R. Lemmert, On universal subsets of Banach spaces, Math. Z.229 (1998), no. 4, 615–619.

[4] A. Peris, Set-valued discrete chaos, Chaos Solitons Fractals 26 (2005), no. 1,19–23.

1Departamento de Matematica Aplicada, Instituto de Matematica, Univer-sidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ,21945-970, [email protected]

2IUMPA, Universitat Politecnica de Valencia, Departament de MatematicaAplicada, Edifici 7A, 46022 Valencia, [email protected], [email protected]

27



My friend Manuel Valdivia

Jean Schmets1,

The aim of this talk is twofold.In the first part I recall a few major facts concerning Professor Manuel Valdivia.The second part is dedicated to my co-author Manuel, the functional analyst I got

to meet, to know as a person, to appreciate as a scholar and who became a very closefriend.

1Department of Mathematics, University of LiegeAllee de la Decouverte, 12Sart TilmanB-4000 LIEGE / [email protected]

28



Isomorphic copies of `1 for m-homogeneousnon-analytic Bohnenblust-Hille polynomials

Juan B. Seoane-Sepulveda1

We employ a classical result by Toeplitz (1913, [3]) and the seminal work byBohnenblust and Hille on Dirichlet series (1931, [1]) to show that the set of continuousm-homogeneous non-analytic polynomials on c0 contains an isomorphic copy of `1.Moreover, we can have this copy of `1 in such a way that every non-zero elementof it fails to be analytic at, precisely, every non-zero point belonging to an infinitedimensional linear subspace.

The material presented in the lecture is part of a joint work ([2]) with J. AlbertoConejero and Pablo Sevilla-Peris (IUMPA - Universitat Politecnica de Valencia,Spain).

References

[1] H. F. Bohnenblust, H. F. and E. Hille, E., On the absolute convergence ofDirichlet series. Ann. of Math. (2) 1931, 600–622.

[2] J. A. Conejero, J.B. Seoane-Sepulveda, and P. Sevilla-Peris, Isomor-phic copies of `1 for m-homogeneous non-analytic Bohnenblust–Hille polynomials.Preprint, 2016.

[3] O. Toeplitz, Uber eine bei den Dirichletschen Reihen auftretende Aufgabe ausder Theorie der Potenzreihen von unendlichvielen Veranderlichen. Nachrichtenvon der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, 1913, 417–432.

1Departamento de Analisis Matematico,Facultad de Ciencias Matematicas,Plaza de Ciencias 3,Universidad Complutense de Madrid,28040 Madrid, Spainand

Instituto de Ciencias Matematicas (CSIC-UAM-UC3M-UCM)C/ Nicolas Cabrera 13–15Campus de Cantoblanco, UAM28049 Madrid, [email protected]

29



On dependently embedded sets

Enrique Casanovas,

Stably embedded sets have been studied in [1]. They help, for example, to un-derstand the relation between stability of a type-definable set and stability of theinduced structure on this set. It turns out that a similar notion (we call it dependentembeddedness) can be used in a NIP context. We will discuss the basic propertiesof dependently embedded sets, with special attention to the uniform version of theproperty. There are strong connections with Chernikov and Simon’s result on honestdefinitions and UDTFS of [2], [3] and [4].

References

[1] Z. Chatzidakis and E. Hrushovski. Model theory of difference fields. Trans-actions of the American Mathematical Society, 351 (1999), 2997–3071.

[2] A. Chernikov and P. Simon. Externally definable sets and dependent pairs.Israel Journal of Mathematics, 194 (2013), 409–425.

[3] A. Chernikov and P. Simon. Externally definable sets and dependent pairs II.Transactions of the American Mathematical Society, 367 (2015), 5217–5235.

[4] P. Simon. A guide to NIP theories. Lecture Notes in Logic 44, CambridgeUniversity Press, Cambridge, 2015.

Department of Mathematics and Computer Science,University of Barcelona,Gran Via 585, 08007 Barcelona, [email protected]

30



P -minimality and p-adic integration

Pablo Cubides Kovacsics1, Eva Leenknegt2,

A celebrated result of Igusa establishes that given a polynomial f(x) ∈ Z[x], its as-sociated Poincare series Pf (T ) is rational. Here Pf (T ) denotes the series

∑m∈N NmTm

whereNm := #x ∈ (Z/pmZ)n | f(x) ≡ 0 mod pm.

In [4], Denef gave an alternative proof for this result by translating the original prob-lem into a problem about p-adic integration. The main part of his strategy consists inshowing that a certain algebra of functions called “constructible functions” is stableunder integration. His ideas were generalized to different classes of functions, obtain-ing rationality results for new Poincare series (cf. [1, 2]). In this talk I will discuss ageneralization of this result to P -minimality, a p-adic analog of o-minimality. This isa joint work with Eva Leenknegt [3].

References

[1] R. Cluckers and F. Loeser, Constructible exponential functions, motivicFourier transform and transfer principle, Ann. Math. 171(2) (2010), 1011–1065.

[2] R. Cluckers, J. Gordon, and I. Halupczok, Integrability of oscillatory func-tions on local fields: transfer principles, Duke Math. J. 136(8) (2014), 1549–1600.

[3] P. Cubides Kovacsics and E. Leenknegt, Integration and cell decompositionin P -minimal structures, To appear in Journal of Symbolic Logic.

[4] J, Denef, The rationality of the Poincare series associated to the p-adic pointson a variety., Invent. Math. 77 (1984), 1–23.

1Laboratoire LMNO, Universite de Caen, BP 5186, F 14032 Caen Cedex,[email protected]

2Department of Mathematics, KULeuven, Celestijnenlaan 200B, 3001 Hever-lee, [email protected], emailthird@address

31



Expansion of valued fields by multiplicativesubgroup

Nathanael Mariaule1

Let K be a field and G be a multiplicative subgroup of K. Can we describeTh(K,G)? In [1], L. van den Dries and A. Gunaydin consider this problem when Kis the field of real or complex numbers and when G has Mann property. It is a naturalquestion to ask if such results can also be achieved for valued fields. In this talk, I willpresent special cases of this problem. First, for K = Qp and G = nZ where n ∈ N.Then, for K = Cp and G is the kernel of the Iwasawa logarithm. In both cases, G hasthe Mann property. I will give an axiomatisation of the theories and describe theirdefinable sets.

References

[1] A. Gunaydin et L. van den Dries,The fields of real and complex numbers with asmall multiplicative group, Proceedings of London Mathematical Society 93, 43-81(2006)

1Universite Paris Diderotmariaule [email protected]

32



A simple remark

Amador Martin-Pizarro

Several theories of pure fields or fields equipped with operators are simple, withnon-forking described purely in terms of the algebraic independence of the structuresin the expanded language. We will provide an abstract unifying approach, whichcovers in particular classical theories, such as differentially and difference closed fields,separably closed fields of finite imperfection degree as well as perfect PAC fields withbounded Galois group.

Charge de Recherche - Institut Camille Jordan, UMR 5208Centre National de la Recherche Scientifique C.N.R.S.43 boulevard 11 novembre 1918 - 69622 Villeurbanne [email protected]

33



Superrosy division rings

Daniel Palacın1

In this talk we analyze superrosy division rings, i.e. division rings which admita well-behaved ordinal valued rank function on definable sets that behaves like arudimentary notion of dimension. Examples are the quaternions, superstable divisionrings (which are known to be algebraically closed fields [2, 1]) and more generallysupersimple division rings which are commutative [3].

In the talk I shall present superrosyness as a common generalization of o-minimalityand supersimplicity and then explain why any superrosy division ring has finite di-mension over its center. This is a joint work with Nadja Hempel.

References

[1] G. Cherlin and S. Shelah, Superstable fields and groups. Ann. Math. Logic18 (1980), 227–270.

[2] A. Macintyre, On ω1-categorical theories of fields. Fund. Math. 71 (1971), 1–25.

[3] A. Pillay, T. Scanlon, and F. O. Wagner, Supersimple fields and divisionrings. Math. Res. Lett. 5 (1998), 473–483.

1Institut fur Mathematische Logik und Grundlagenforschung, UniversitatMunster, Einsteinstrasse 62, 48149 [email protected]

34



Oscillatory integrals and o-minimality

T. Servi, R. Cluckers, G. Comte, D. Miller, J.P. Rolin

We prove the stability under integration and under Fourier transform of a concreteclass E of functions, containing all globally subanalytic functions and their complexexponentials. The class E is a system of algebras of which we describe explicitly thegenerators. The methods of proof pertain to o-minimality (in particular, subanalyticresolution of singularities) and to the theory of almost periodic functions. This pro-vides an example of a fruitful interaction between singularity theory, o-minimalityand analysis.

University Paris [email protected]

35



Geometric constructions as a transfer tool inTropical Geometry

Luis Felipe Tabera Alonso1

Given an algebraically closed field provided with a valuation, tropical geometrycan be seen as doing geometry on the value group alone. We will show a collection ofsituations that show the difference between classical and tropical geometry. The veryfirst example is that, by two different points, there may be infinitely many distinctlines passing through them.

Equivalent theorems in classical geometry may not be equivalent in tropical ge-ometry. But, if a theorem can be described by a geometric construction with certainproperties, then if it is true in classical geometry, then it will be true in tropicalgeometry. We will also show that, in certain sense, tropical geometry must inheritproperties from algebraic geometry over any characteristic.

References

[1] D. Speyer, B. Sturmfels, The Tropical Grassmannian, Advances in Geometryvolume 4, (2004), 389–411.

[2] L. F. Tabera, Tropical Plane Geometric Constructions: a Transfer Techniquein Tropical Geometry, Revista Matemtica Iberoamericana volume 27, (2011),181–232.

1Departamento de Matematicas, Estadıstica y Computacion,Universidad de Cantabria,Av. Los Castros s/n 39005, Santander, Spain

[email protected]

36



On generalized Goodstein sequences

Andreas Weiermann1,

The termination property of the classical Goodstein sequences provides a simplenumber-theoretic assertion which is true but independent of first order Peano arith-metic PA.

In this talk we consider Goodstein sequences which are defined relative to Acker-mannian functions. We discuss the following two results.

Theorem A. When the zero-th branch of the Ackermann function is the successorfunction then the induced Goodstein principle will be equivalent to the one consistencyof PA.

Theorem B. When the zero-th branch of the Ackermann function is an exponen-tial function k 7→ kb then the induced Goodstein principle will be equivalent to theone consistency of ATR0.

(Theorem B is joint work with Tosiyasu Arai and Stan Wainer). If time is left weconsider the strength of Goodstein principles relative to the fast growing Schwichten-berg Wainer hierarchy.

1Department for Mathematics Ghent University, Krijgslaan 281 Building [email protected]

37



Artin and Hilbert type theorems for Lie algebras

Ana Agore1

If g ⊆ h is an extension of Lie algebras over a field k such that dimk(g) = n anddimk(h) = n+m, then the Galois group Gal (h/g) is explicitly described as a subgroupof the canonical semidirect product of groups GL(m, k) o Mn×m(k). An Artin typetheorem for Lie algebras is proved: if a group G whose order is invertible in k actsas automorphisms on a Lie algebra h, then h is isomorphic to a skew crossed producthG #• V , where hG is the subalgebra of invariants and V is the kernel of the Reynoldsoperator. The Galois group Gal (h/hG) is also computed, highlighting the differencefrom the classical Galois theory of fields where the corresponding group is G. Thecounterpart for Lie algebras of Hilbert’s Theorem 90 is proved and based on it thestructure of Lie algebras h having a certain type of action of a finite cyclic group isdescribed. Radical extensions of finite dimensional Lie algebras are introduced and itis shown that their Galois group is solvable. Several applications and examples areprovided. Based on a joint work with G. Militaru ([1]).

References

[1] A. Agore and G. Militaru, Galois groups and group actions on Lie algebras,preprint arXiv:1505.07346

1Vrije Universiteit Brussel, Pleinlaan 2 B-1050 Brussel - [email protected]

38



Gradings on exceptional simple Jordan systems andstructurable algebras

Diego Aranda-Orna1

This talk is about some generalities of group gradings on structurable algebras andKantor systems. We explain how these gradings can be used to construct gradingson the associated Lie algebra given by the Kantor construction.

Some results of classification of equivalence classes of fine gradings by abeliangroups are given for exceptional simple Jordan pairs and triple systems (the onesof types bi-Cayley and Albert). Also, an example of fine Z3

4-grading on the Brownalgebra (a 56-dimensional simple structurable algebra) is given. We explain how someof these gradings can be used to construct fine gradings on some exceptional simpleLie algebras.

1Departamento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza,[email protected]

39



On bialgebroids and warpings

Mitchell Buckley1, Joost Vercruysse2

There are many different characterisations of bialgebras and their generalisations.Each characterisation lends itself to different weaker notions of bialgebra and alter-native conditions for identifying Hopf algebras among bialgebras. In particular, abialgebroid A over base algebra R can be characterised as:

1. an opmonoidal monad on RModR possessing a right adjoint [1, 2];

2. a closed right skew monoidal structure on ModR with unit R [3]; or

3. a monoidal structure on ModA which is strictly preserved by the forgetful functorto RModR [4].

It is not trivial to show that these are equivalent, yet the underlying relationshipsfollow from abstract arguments concerning monads, warpings, and skew monoidalcategories.

Without reproducing the above equivalences entirely, we show how the bases ofthese results rely on simple categorical constructions. In particular we highlight therole of warpings and their relationship with fusion operators and opmonoidal monads.This work is still in progress.

References

[1] K. Szlachanyi, The monoidal Eilenberg-Moore construction and bialgebroids,Journal of Pure and Applied Algebra, 182(2–3) (2003), 287–315.

[2] G. Bohm, Hopf algebroids, in Handbook of algebra. Vol. 6, Elsevier, Amsterdam,2009.

[3] K. Szlachanyi, Skew-monoidal categories and bialgebroids, Advances in Math-ematics, 231(3–4), (2012), 1694 – 1730.

[4] P. Schauenburg, Bialgebras over noncommutative rings and a structure theoremfor Hopf bimodules, Appl. Categ. Structures, 6(2) (1998), 193–222.

1Departement de Mathematiques, Faculte des sciences, Universite Libre deBruxelles, Boulevard du Triomphe, B-1050 Bruxelles, [email protected]

2Departement de Mathematiques, Faculte des sciences, Universite Libre deBruxelles, Boulevard du Triomphe, B-1050 Bruxelles, [email protected]

40



On the Prime Graph Question for symmetric groups

M. Caicedo1, A. Bachle2,

Given a finite group G, the integral group ring ZG of G over the integers Z, isdefined as the ring of all linear combinations of the form u =

∑g∈G ugg, where the

coefficients ug are integers. The addition is defined componentwise and the multipli-cation is the extension of the group multiplication. Let U1(ZG) denote the group ofnormalized units of ZG. There are many interesting questions around such a groupof units, specially for units of finite order. An example of this is the Prime GraphQuestion. The prime graph Γ(G) of a group G, which was introduced by Gruenbergand Kegel, is the graph having as vertices those rational primes p for which thereexists an element of this order in the group G and two vertices p and q are connectedby an edge if there is an element of order pq in G. Then the Prime Graph Questionasks whether Γ(U1(ZG)) = Γ(G) for a finite group G.

Let G be an almost simple groups with socle An, the alternating group of degreen. We prove that there is a unit of order pq in the integral group ring of G if andonly if there is an element of that order in G provided p and q are primes greaterthan n

3 . We combine this with some explicit computations to verify the Prime GraphQuestion for all almost simple groups with socle An if n ≤ 17.

1Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050Brussels, [email protected]

2Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050Brussels, [email protected]

41



Units in Integral Group Rings via FundamentalDomains and Hyperbolic Geometry

Ann Kiefer1,

The motivation of this work is the investigation on the unit group of an integralgroup ring U(ZG) for a finite group G. By the Wedderburn-Artin Theorem, the studyof U(ZG) may be reduced, up to commensurability, to the study of SLn(O) for n ≥ 1and O an order in some division ring D. There exists descriptions of a finite setof generators for a subgroup of finite index in SLn(O) for a large number of cases.Excluded from this result are the so-called exceptional components of QG.

Our work consists in finding a presentation, for SLn(O) associated to some ofthese exceptional components. In all the cases we treat, the group SLn(O) has adiscontinuous action on hyperbolic space of dimension 2 or 3, on hyperbolic spaceof higher dimensions, or on some product of hyperbolic spaces. By constructingfundamental domains for these discontinuous actions, we get generators for the groupsin question.

1Faculty of Mathematics Bielefeld University Postfach 100131 D-33501 Biele-feld [email protected]

42



Weak wreath products andweak quantum duplicates

Esperanza Lopez Centella1

Twisted tensor products (wreath products) are the key for the (strict) factorizationproblem of unital algebras. As it is well-known, for a unital algebra, there is a one-to-one correspondence between the set of factorization structures admitting two givenalgebras as factors and the set of so-called twisting maps (distributive laws). Inthis talk we recall the factorization problem answered by a weak wreath product ofalgebras. Motivated by arguments from Mathematical Physics, we introduce thenotion of weak quantum duplicate of an algebra, a construction based on a weakwreath product of the algebra under consideration and a two-dimensional factor. Weprovide a characterization of weak quantum duplicates of a finite-dimensional algebra,extending that one of quantum duplicates given in [1]. As an application, we explicitlydescribe a great part of the set of weak factorization structures (and weak distributivelaws [3]) existing between two two-dimensional unital algebras over a field, classifying(up-to isomorphism) the weak wreath products arising from them and covering thedescription in [2].

References

[1] C. Cibils, Non-conmutative duplicates of finite sets, J. Algebra Appl. 5 (2006),no. 3, 361-377.

[2] O. Cortadellas, G. Navarro, J. Lopez-Pena, Factorizations structures witha two-dimensional factor, J. London Math. Soc 2 (2010), no. 81, 1-23.

[3] R. Street, Weak distributive laws, Theory and Appl. of Categories 22 (2009),no. 22, 313–320.

1Departmento de Algebra, Universidad de Granada, Facultad de Ciencias,Avenida de Fuente Nueva s/n, 18071 Granada (Spain)[email protected]

43



Darmon points on modular abelian varieties overtotally real fields

Xevi Guitart1, Marc Masdeu2, Santiago Molina3

Let A be a modular abelian variety over a totally real field F and let K/F be aquadratic extension. If K/F is totally imaginary, the theory of complex multiplicationprovides a construction of certain special points on A. These so-called Heegner pointsgive rise to partial approaches to the Birch and Swinnerton-Dyer conjecture.

In this talk, we present new conjectural constructions of special points on A at-tached to non-CM extensions K/F . Such constructions generalize [1], [2], [3], [4] andthe classical construction of Heegner points. We predict the Galois action on suchpoints, called Stark-Heegner or Darmon points, and its connection with the Birchand Swinnerton-Dyer conjecture. Our conjectures are supported by many numericalevidence.

References

[1] H. Darmon, Integration on Hp × H and arithmetic applications. Ann. of Math.(2), 154(3):589639, 2001

[2] H. Darmon, Rational points on modular elliptic curves, volume 101 of CBMSRegional Conference Series in Mathematics. Published for the Conference Boardof the Mathematical Sciences, Washington, DC, 2004.

[3] x. Guitart, M. Masdeu, M.H. Sengun, Darmon points on elliptic curves overnumber fields of arbitrary signature. To appear in P. Lond. Math. Soc.

[4] J. Gartner, Darmon’s points and quaternionic Shimura varieties. In Canad. J.Math., 64(6):12481288, 2012.

1Department de mateatiques, Universitat de Barcelona, Gran via de les CortsCatalanes, 585 08007 Barcelona (Spain)[email protected]

2Mathematics Department, University of Warwick, Warwick Mathematics In-stitute Zeeman Building Coventry, CV4 7AL (United Kingdoms)[email protected]

3Centre de Recerca Matematica, Campus de Bellaterra, Edifici C 08193 Bel-laterra (Spain)[email protected]

44



Homological torsionoriginating from finite subgroups

Alexander D. Rahm

For calculating the part of the torsion in the homology of an arithmetic groupwhich is induced by its finite subgroups, some ad-hoc tricks have been known toexperts [4], but those tricks did for decades not get explained in published form.The speaker did rediscover these tricks, and organise them into a technique, TorsionSubcomplex Reduction [2]. The advantage of using this systematic technique ratherthan a set of ad-hoc tricks, is that one is no more limited to example calculations,but can develop number-theoretic formulas which hold for whole classes of arithmeticgroups [1]. This sort of formulas is fuelling the recent progress on the Quillen–Wendtconjecture [3].

References

[1] A. D. Rahm, Accessing the cohomology of discrete groups above their virtualcohomological dimension, J. Algebra 404 (2014), 152–175

[2] A. D. Rahm, The homological torsion of PSL2 of the imaginary quadratic inte-gers, Trans. Amer. Math. Soc. 365 (2013), 1603–1635

[3] A. D. Rahm and M. Wendt, A refinement of a conjecture of Quillen, C. R.Math. Acad. Sci. Paris 353 (2015), 779-784

[4] C. Soule, The cohomology of SL3(Z), Topology 17 (1978), 1–22

Mathematics Research Unit, Universite du Luxembourg, 6 rue RichardCoudenhove-Kalergi, L-1359 [email protected]

45



Nonmatrix varieties for some classes of nonassociative algebras

Ivan Shestakov

A variety M of associative algebras (over a field F ) is called “nonmatrix” ifF2 /∈ M , where F2 is the usual matrix algebra of second order over F . V.N. Latyshevintroduced these varieties in 1977. Concerning this definition, other equivalent char-acterizations for a nonmatrix variety were obtained, for instance, by considering al-gebraic (G. Chekanu, 1979) and nilpotent (A. Mishchenko et al, 2012) elements. Non-matrix varieties are studied mainly in the case of characteristic zero for associativealgebras.

However, the theory of varieties of algebras is not restricted to the class of associa-tive algebras. In addition to the Lie algebras, among many classes of non associativealgebras, we highlight the alternative, the Jordan and the non commutative Jordanalgebras. These classes of algebras have many connections and applications to severalareas of Mathematics and Physics and have a well-developed structural theory, as inthe class of associative algebras.

The concept of “nonmatrix variety” can be reformulated in the classes of algebrasabove and our work is to adapt, extend or generalize some results, as mentioned, fornon-matrix varieties in these classes of algebras.

Universidade de Sao [email protected]

46



The self-normalizing case of the McKay conjecture

Carolina Vallejo Rodrıguez

Let p be a prime number and let G be a finite group. The McKay conjectureasserts that there exists a bijection between the irreducible characters of degree notdivisible by p of G and those of the normalizer of a Sylow p-subgroup P of G.

We will study some special cases in which not only can we find a bijection betweenthese two sets but a natural one. By natural we mean that an algorithm to computethe bijection is provided and the result does not depend on the choices made in itsapplication.

Departament d’Algebra, Universitat de ValenciaDr. Moliner 50, 46100, Burjassot, [email protected]

47



On the Distribution of Frobenius of Weight 2Eigenforms with Quadratic Coefficient Field

J. Van Hirtum

The coefficients of a modular form without so called inner twists are elements of acertain totally real number field. If this number field is different from Q then one canstudy the set of primes p such that the p-th coefficient is a rational number. This setis known to be of density zero. However only conjectural statements exists on its size.Using the latest results on the Sato-Tate conjecture for abelian varieties we obtaina heuristic model for the asymptotic size of this set under reasonable assumptions.More precisely we treat the case of weight 2 eigenforms with quadratic coefficientfield without inner twist. Moreover we present numerical data which agrees with ourmodel and the assumptions we made to obtain it.

Department of Mathemetics, KU Leuven,Celestijnenlaan 200 B, B-3001 Heverlee, BelgiumMathematics Research Unit FSTC, Universite du Luxembourg,6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, [email protected]

48



Standing wave solutions for a nonlinear Schrdingerequation with mixed dispersion

Jean-Baptiste Casteras1,

In this talk, we will be interested in standing wave solutions to a fourth ordernonlinear Schrdinger equation having second and fourth order dispersion terms. Thiskind of equation naturally appears in nonlinear optics. In a first time, we will establishthe existence of ground-state solutions. We can obtain them with two different ways: either by imposing a L2 mass constraint or via a Lp constraint. We will then beinterested in their qualitative properties : positivity, symmetry, exponential decay,uniqueness and orbital stability. We will conclude with some open questions. Jointwork with Denis Bonheure, Ederson Moreira Dos Santos and Robson Nascimento.

1Universite Libre de [email protected]

49



Nonlinear elliptic singular systems with quadraticgradient lower order terms

Jose Carmona Tapia1,

We present some results obtained in [2] on the existence of solution for quasilinearelliptic systems with lower order terms having quadratic growth in the gradient andsingularities. The model problem is the following:

−∆u+ g1(v)|∇u|2uα

= f1(x, u, v) in Ω,

−∆v + g2(u)|∇v|2vβ

= f2(x, u, v) in Ω,

u, v ∈ H10 (Ω),

(1)

where Ω ⊂ RN is a smooth bounded domain, 0 < α, β < 1 and f1, f2, g1, g2 areregular functions. A recent comparison principle for the scalar equation in [1] allowus to prove a sub-supersolution method and to deal with classical Lotka-Volterramodels.

References

[1] D. Arcoya, J. Carmona and P. J. Martınez-Aparicio, Comparison prin-ciple for elliptic equations in divergence with singular lower order terms havingnatural growth. Commun. Contemp. Math. DOI: 10.1142/S0219199716500139.

[2] J. Carmona, P. J. Martınez-Aparicio and A. Suarez, A sub-supersolutionmethod for Nonlinear elliptic singular systems with natural growth and someapplications. Nonlinear Analysis 132 (2016), 47-65.

1Department of Mathematics, University of Almerıa, Ctra. Sacramento sn,04120, La Canada de San Urbano, Almerı[email protected]

50



Liouville type equations with sign-change data

Rafael Lopez Soriano1,

In this talk, we study the existence of solutions of the following problem

−∆gu = λ

(Keu∫

ΣKeudVg

− 1

)in Σ (1)

where Σ is a compact surface without boundary, equipped with a Riemannian metricg, ∆g is the Laplace-Beltrami operator, λ > 0, V olg(Σ) =

∫Σ

1dVg = 1 and K is afunction defined in Σ. It is well-known that (1) has a geometric meaning, arising inthe assigned Gauss curvature problem. From a variational point of view, the problemhas been extensively studied, see for example [4], for the regular case, and [1] for thesingular one, under the assumption that K is positive.

However, imposing additional hypotheses on∇K in the region where K = 0, basedof a moving-planes argument and a priori estimates, it is possible give a existenceresult for the sign-changing case for the sphere, [2], and for general surfaces [3].

References

[1] D. Bartolucci, F. De Marchis, A. Malchiodi, Supercritical conformal metrics withconical singularities, Int. Mat. Res. Not. 24 (2011), pp. 5625–5643.

[2] F. De Marchis, R. Lopez-Soriano Existence and Non Existence Results for thesingular Nirenberg problem, Calc. Var. Partial Differential Equations, 55 (2016),no. 2, 55:36.

[3] F. De Marchis, R. Lopez-Soriano work in progress.

[4] Z. Djadli, Existence result for the mean field problem on Riemann surfaces of allgenuses, Commun. Contemp. Math. 10 (2008), no. 2, 205–220.

1Universidad de [email protected]

51



On the eigenvalues of Aharonov-Bohm operatorswith varying poles

Manon Nys1,

Let Ω be an open, simply connected set in R2. For a = (a1, a2) ∈ Ω, we considerthe following magnetic operator: (i∇+Aa)2 acting on functions u with zero boundaryconditions on ∂Ω. The magnetic potential Aa, singular at the point a and havingcirculation α around a, has the following form

Aa(x1, x2) = α

(− x2 − a2|x− a|2 ,

x1 − a1|x− a|2

).

It corresponds to a magnetic field, being a multiple of the delta-Dirac, orthogonal toΩ at a. Those two objects are related to the Bohm-Aharonov effect. More partic-ularly, we are interested in the spectrum of this operator when the pole a moves inΩ and eventually approaches the boundary of Ω. For all circulation α ∈ (0, 1), wehave proved the continuity of the magnetic eigenvalue with respect to a; and differ-entiability as long as the eigenvalue remains simple. We also prove the continuityup to the boundary of Ω: as a converges to the boundary of Ω, the k-th magneticeigenvalue converges to the k-th eigenvalue of the Laplacian with Dirichlet boundaryconditions. This implies in particular that the magnetic eigenvalue (as a function ofa) always has an extremal point in Ω. In the case of half-integer circulation α = 1/2,this problem is related to a Laplacian problem in a double covering. For this reason,we can deduce the behaviour of the nodal lines of the magnetic eigenfunctions: faraway from the singularity a, the eigenfunction behaves like an eigenfunction of theLaplacian, while at the singular point a, it has an odd number of nodal lines. Then,we study additional properties of extremal interior points. We will show that the rateof convergence of the magnetic eigenvalues when a goes to some fix point b dependson the number of nodal lines of the corresponding magnetic eigenfunction with poleat b. For this, we use inversion theorem. We can also prove a rate of convergence ofthe eigenvalues when the pole a approaches the boundary. This requires completelydifferent techniques. We use then an Algrem?s type formula. Those results are cor-roborated by numerical simulations in the case when Ω is an angular sector or asquare.Joint work with Virginie Bonnaillie-Noel, Benedetta Noris and Susanna Terracini.

1Universit degli Studi di [email protected]

52



On the extended Allen-Cahn equation

Alberto Saldana1,

Nonlinear fourth-order PDEs usually have a richer and more complex set of so-lutions when compared to its second-order counterpart. In this sense, many modelsexhibit behaviors that could be better described with fourth-order equations, likeocean and atmosphere dynamics, bridges, and pattern formation, just to mentionsome of them. The theory for higher-order nonlinear problems, however, is far lessdeveloped than its second-order analogue and many basic questions remain open.Lack of maximum principles, oscillatory behavior of solutions, and regularity issuesare some of the main difficulties in the study of such problems.

In this talk I consider a fourth-order extension of the Allen-Cahn model withmixed-diffusion and Navier boundary conditions. I present results on existence,uniqueness, positivity, stability, a priori estimates, and symmetry. As an application,we construct a saddle solution in the whole space. The proofs rely on variationaland bifurcation methods. Some numerical approximations of solutions will also bediscussed.

1Universite Libre de [email protected]

53



Some problems arising in population dynamics withnon-linear diffusion

Antonio Suarez1,

In this talk we present some theoretical results about nonlinear partial differentialequations coming from population dynamics. Specifically, we focus our attention inmodels where the diffusion (the spatial movement of the species) is non-linear; that is,it depends on the value of the species in a non-linear way or even depends on anotherspecies living in the habitat. We show, through concrete examples, the main differ-ences between linear diffusion models and nonlinear diffusion and the mathematicaldifficulties arising in this case.

1Universidad de [email protected]

54



Constancy regions of mixed multiplier ideals inrational surface singularities

Marıa Alberich Carraminana1

We will present some new results on the computation of jumping walls of anyr-tuple of ideal sheafs around a rational surface singularity. More precisely, we willfocus on an iterative method to compute all the constancy regions (giving the samemixed multiplier ideal) and their boundary (the jumping walls) in any compact regionof the positive orthant. We provide also a characterization whether a given point liesin a jumping wall. This is a joint work with J. lvarez Montaner and F.Dachs-Cadefau.

1Departament de Matematica Aplicada IETSEIB-UPCAvda. Diagonal 64708028 BarcelonaMaria.Alberichupc.edu

55



Geometric invariants encoded in the Newtonpolygon

Wouter Castryck1

Let C be the plane curve defined by a bivariate Laurent polynomial f that issufficiently generic (in some precise sense) with respect to its Newton polygon ∆(f).It was proven by Khovanskii in 1977 that the geometric genus of C equals the numberof points in the interior of ∆(f) that have integral coordinates. In this talk I willgive a list of other invariants that allow for a similar combinatorial interpretation, themost notable of which are the gonality and the Clifford index. This will include jointwork with Filip Cools, Jeroen Demeyer and Alexander Lemmens.

1Universiteit Gent - Vakgroep WiskundeGebouw S22, Krijgslaan 2819000 Gent, [email protected]

56



Splicing and zeta functions

Thomas Cauwbergs1

Many properties of plane curve singularities are contained in the so-called splicediagram. By incorporating a differential form into the splice diagram, Nmethi andVeys proved a splicing formula. This splice diagram is essentially a decorated dualgraph of an embedded resolution and splicing is operation on these splice diagrams.It splits such a graph into two parts and the involved topological zeta functionsare related by this splicing formula. An interesting question is then what happensif we look at more general zeta functions such as the motivic zeta function and themonodromic motivic zeta functions. I discuss these (splice) diagrams and give anotherproof of the splicing formula. The advantage of this proof is that it also is valid forthese other zeta functions. However I will also discuss some problems arising fromconsidering these other zeta functions.

1University of [email protected]

57



Topology of spaces of valuations and geometry ofsingularities

Ana Belen De Felipe1

Given an algebraic variety X defined over a field k, the space of all valuations ofthe field of rational functions of X extending the trivial valuation on k is a projectivelimit of algebraic varieties. This space had an important role in the program of Zariskifor the proof of the existence of resolution of singularities.

In this talk we will consider the subspace RZ(X,x) consisting of those valuationswhich are centered in a given closed point x of X and we will focus on the topology ofthis space. In particular we will concentrate on the relation between its homeomor-phism type and the local geometry of X at x. We will characterize this homeomor-phism type for regular points and normal surface singularities. This will be done bystudying the relation between RZ(X,x) and the normalized non-Archimedean link ofx in X coming from the point of view of Berkovich geometry.

1Universidad de La Laguna / [email protected]

58



The arc space of the Grassmannian

Roi Docampo1

Arc spaces can be used as an effective tool to compute invariants of singularitiesof algebraic varieties. In this talk, I will explain how this can be achieved for aclassical example: the singularities of Schubert varieties inside the Grassmannian.This involves a delicate study of the combinatorics inside of the arc space of theGrassmannian. The main tool I will discuss is a stratification of the arc space whichplays the role of a Schubert cell decomposition for lattices. Analyzing the geometricstructure of the resulting strata leads to the computation of invariants, mainly thelog canonical threshold of pairs involving Shubert varieties.

[email protected]

59



Nearby cycles and Alexander modules ofhypersurface complements

Yongqiang Liu1

Assume that f is a reduced polynomial and generic at infinity. Then f=0 definesa hypersurface. Using an explicit construction, we show that the Alexander modulesof hypersurface complements (up to middle degree) are torsion, semisimple, and theyhave mixed Hodge structures. Moreover, we realize these Alexander modules bynearby cycles. As an application, we give the divisibility results for these Alexandermodules.

1KU [email protected]

60



On the 2-Jordan blocks for the eigenvalue λ = 1 ofisolated singularities

Jorge Martın-Morales1

In the isolated case, Steenbrink provided a spectral sequence converging to thecohomology of the Milnor fiber using the semistable reduction associated with anembedded resolution of the singularity. During the talk, we will study the sameproblem starting with abelian quotient singularities in the ambient space and we willapply it to certain surface singularities for the eigenvalue 1.

1Centro Universitario DefensaCtra. de Huesca s/n.50090 Zaragoza, [email protected]

61



On the estimation of numerical invariants of agraded module in terms of its Hilbert series

Julio Jose Moyano-Fernandez1

Let M be a graded module over a polynomial ring, and H(M) the Hilbert seriesof M . It is interesting to estimate numerical invariants of M which cannot be easilyread off from H(M), such as the depth and the Castelnuovo-Mumford regularity. Inthis talk we will explain the main ingredients in our algebraic-combinatorial approachto the latter invariants, based on the notions of Hilbert depth and Hilbert regularity.This is (part of) several joint works with W. Bruns, B. Ichim, L. Katthn and J.Uliczka.

1Universitat Jaume [email protected]

62



Positivity of divisors on blown-up projective spaces

Elisa Postinghel1

The minimal model program aims at a birational classification of algebraic vari-eties. The abundance conjecture and the existence of good models are among themain open problems in this field. In this talk we study log canonical pairs given bydivisors on blow-ups of projective spaces. Vanishing theorems for the higher cohomol-ogy groups are used to give a systematic study of semi-ampleness and other positivityproperties of these divisors. These imply a proof of the abundance conjecture for thecorresponding pairs, and an explicit construction of good minimal models.?This isjoint work with Olivia Dumitrescu.

1KU [email protected]

63



Duality on value semigroups

Mathias Schulze1

The semigroup of values is a classical combinatorial invariant associated to a curvesingularity. It is defined by taking all regular elements of the corresponding ring to theintegral closure and taking a multivaluation. For a curve singularity with r branchesthe semigroup of values is a submonoid of Nr. In the irreducible case it is a numericalsemigroup, otherwise it is not even finitely generated. Due to Lejeune-Jalabert andZariski, this value semigroup determines the topological type of plane complex curves.As observed by Kunz in the irreducible case, the Gorenstein property of a curvesingularity is equivalent to a symmetry of gaps and non-gaps in the (numerical) valuesemigroup. Delgado generalized this result to the reducible case introducing a non-obvious notion of symmetry of a semigroup. A canonical (fractional) ideal on acurve singularity defines a duality on fractional ideals. On the other hand takingmultivaluations as above associates to any fractional ideal a value semigroup ideal.Such value semigroup ideals satisfy certain natural axioms defining the class of so-called good semigroup ideals. Barucci, D’Anna and Frberg gave an example of a goodsemigroup that does not come from a ring. Extending Delgado’s symmetry result,D’Anna described the value semigroup ideals of canonical ideals. In the Gorensteincase, Delphine Pol described the value semigroup ideal of duals. Unifying the workof D’Anna and Pol we establish a purely combinatorial duality on good semigroupideals that mirrors the duality on fractional ideals. The talk is based on joint workwith Philipp Korell and Laura Tozzo.

1Technische Universitat [email protected]

64



A family of symplectic-complex Calabi-Yaumanifolds that are nonKahler

Botong Wang1

A Kahler manifold is a smooth manifold with compatible complex and symplecticstructures. In general, a compact manifold which admits both complex and symplecticstructures may not admit any Kahler structure. Hodge theory and hard Lefschetztheorem have very strong implications on the homotopy type of compact Kahlermanifolds. We introduce a family of 6-dimensional compact manifolds M(A), whichadmit both Calabi-Yau symplectic and Calabi-Yau complex structures. They satisfyall the consequences of classical Hodge theory and hard Lefschetz theorem. However,we show that they are not homotopy equivalent to any compact Kahler manifold usinga recently developed cohomology jump loci method. This is joint work with LizhenQin.

1Department of MathematicsUniversity of Wisconsin - [email protected]

65



Coexistence of hyperchaos and chaos: aComputer-assisted proof

Roberto Barrio1, M. Angeles Martınez2, Sergio Serrano1, Daniel Wilczak3

It has recently been reported that it is quite difficult to distinguish between chaosand hyperchaos in numerical simulations which are frequently “noisy”. In this talk weshow that, for the classical 4D Rossler model, the coexistence of two invariant sets withdifferent nature (a global hyperchaotic invariant set and a chaotic attractor) and thehomoclinic and heteroclinic connections between their unstable periodic orbits giverise to long hyperchaotic transient behavior, and therefore it provides a mechanismfor noisy simulations [1].

Moreover, the existence of several hyperchaotic sets provides an explanation of thesmooth change from chaotic to hyperchaotic attractors due to the appearance of newheteroclinic connections among them, and so the joining of the different sets givesrise to slightly bigger and slightly more hyperchaotic attractors in the sense that thesecond Lyapunov exponent grows a little. The same phenomena are expected in other4D and higher dimensional systems.

The Computer-assisted proof of this coexistence of chaotic and hyperchaotic be-haviors combines topological and smooth methods with rigorous numerical computa-tions [2]. The existence of (hyper)chaotic sets is proved by the method of coveringrelations and cone conditions [3].

References

[1] R. Barrio, M. A. Martınez, S. Serrano, and D. Wilczak, When chaosmeets hyperchaos: 4D Rossler model, Phys. Lett. A 379 (2015), 2300–2305.

[2] D. Wilczak, S. Serrano, and R. Barrio, Coexistence and dynamical con-nections between hyperchaos and chaos in the 4D Rossler system: a Computer-assisted proof, SIAM J. Applied Dynamical Systems 15 (2016), 356-390.

[3] H. Kokubu, D. Wilczak, and P. Zgliczynski, Rigorous verification of cocoonbifurcations in the Michelson system, Nonlinearity 20 (2007), 2147.

1Departamento de Matematica Aplicada and IUMA. CoDy group. Universityof Zaragoza, 50009 Zaragoza, (Spain)[email protected], [email protected]

2BSICoS and CoDy groups. CIBER–BBN, 50018 Zaragoza, (Spain)[email protected]

3Faculty of Mathematics and Computer Science. Jagiellonian University. Lojasiewicza 6, 30-348 Krakow, (Poland)[email protected]

66



A journey in the zoo of Turing patterns

Timoteo Carletti1

Self-organizing phenomena are widespread in Nature and have been studied for along time in various domains, be it in physics, chemistry, biology, ecology, neurophys-iology, to name a few [1]. Despite the rich literature on the subject, there is still needfor understanding, analyzing and predicting their behaviors.

They are commonly based on local interaction rules which determine the creationand destruction of the entities at every place, upon which a diffusion process deter-mines the migration of the components. For this reason reaction-diffusion systemsare a common framework of modeling such systems [2].

In a 1952 article in biomathematics, Turing considered a two-species model ofmorphogenesis [3]. For the first time, he established the conditions for a stable spa-tially homogeneous state, to migrate towards a new heterogeneous, spatially patched,equilibrium under the driving effect of diffusion, at odd with the idea that diffu-sion is a source of homogeneity. Even though the explanation for morphogenesis hasevolved and now relies more on genetic programming, many actual results are basedor inspired form this pioneering work. The so-called Turing instabilities, or Turingpatterns, help explain by a simple means the emergence of self-organized collectivepatterns.

The geometry of the underlying support where the reaction-diffusion acts, plays arelevant role in the patterned outcome, it can be because of the non flat geometry [4](possibly growing) [5] or because of its anisotropy [6]. Pushing to the extreme thediscreteness of the space, scholars have considered reaction-diffusion systems on com-plex networks; reactions occur at each node and then products are displaced acrossthe network using the available links, thus possibly exhibiting Turing patterns [7].

The aim of this talk will be to introduce some of the recent developments thatimprove the classical ones by Turing to the framework of more general complex net-works supports, for instance multiplex [8, 9] and cartesian product networks [10], orwith more generic reaction parts, e.g. involving delays.

References

[1] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems:From Dissipative Structures to Order through Fluctuations, Wiley (1977).

[2] P. Grindrod, Patterns and waves: The theory and applications of reaction-diffusion equations, Clarendon Press Oxford, (1991).

[3] A. Turing, The Chemical Basis of Morphogenesis, Phils Trans R Soc LondonSer B, 237, (1952), 37.

[4] C. Varea, J.L. Aragon and R.A. Barrio, Turing patterns on a sphere,Phys. Rev. E, 60, (1999), 4588.

67

[5] R.G. Plaza et al, The Effect of Growth and Curvature on Pattern Formation,J. of Dynamics and Differential Equations, 16, (4), (2004), 1093.

[6] D.M. Busiello et al, Pattern formation for reactive species undergoinganisotropic diffusion, Eur. Phys. J. B, 88, (2015), 88.

[7] H. Nakao and A.S. Mikhailov, Turing patterns in network-organizedactivator-inhibitor systems, Nature Physics, 6, (2010), 544.

[8] M. Asllani et al., Turing patterns in network-organized activator-inhibitorsystems, Physical Review E, 90, (2014), 042814.

[9] N. E. Kouvaris, S. Hata, A. Dıaz-Guilera, Pattern formation in multiplexnetworks, Scientific Report, 5, (2015), 10840.

[10] M. Asllani et al., Turing instabilities on Cartesian product networks,ScientificReport, 5, (2015), 12927.

[11] J. Petit, T. Carletti, M. Asllani and D. Fanelli, Delay-induced Turing-like waves for one-species reaction-diffusion model on a network, European Phys-ical Letter, 111, (2015), 58002

1Department of mathematics and Namur Center for Complex Systems - naXysUniversity of Namur, 8 rempart de la vierge, B5000 [email protected]

68



Space Debris in the Geostationary region as aDynamical System

Daniel Casanova1,2

Satellites play an important role in daily life and provide essential information ona vast number of areas such as Earth observation, telecommunications or navigation.Nowadays, more than 7,437 satellites have been launched into an Earth orbit sinceSputnik I in October 1957.

The location of new satellites in the Geostationary region is a difficult task sincea large quantity of active satellites are placed in this area. In addition, this regioncontains large quantities of space debris, which increases the cost of the satellites toprotect them from the small pieces of debris.

Consequently, time evolution of space debris has recently become a trending topicto predict safer regions and graveyard locations in space. In particular, a piece ofspace debris can be considered as a dynamical system whose equations are:

r = v,

v = akepl + aJ2+ aSRP + a3BS + a3BM .

where r and v represent the position and velocity vectors respectively, akepl accountsfor the attraction of the Earth as a central body, aJ2

accounts for the Earth oblateness,aSRP represents the Solar Radiation Pressure (SRP) acceleration, and finally a3BS

and a3BM include the perturbation due to the Sun and Moon as third bodies.

In this work, an analytical way to propagate space debris in the Geostationaryregion (cf. [1]) is presented. The analytical evolution is compared with a numericalintegrator to prove its efficiency. Furthermore, it is shown the evolution of spacedebris depending on its area to mass ratio (A/m). Finally, potential applications ofthis analytical tool will be presented.

References

[1] D. Casanova, A. Petit, and A. Lemaitre, Long-term evolution of space de-bris under the J2 effect, the solar radiation pressure and the solar and lunar pertur-bations, Celestial Mechanics and Dynamical Astronomy 123(2) (2015), 223–238.

1Area de Matematicas, Estadıstica e Investigacion Operativa. Centro Univer-sitario de la Defensa, Carretera de Huesca s/n. 50090 Zaragoza (Spain)[email protected]

2GME-IUMA, University of Zaragoza, Pedro Cerbuna, 12. 50009 Zaragoza(Spain)

69



Slow-fast Bogdanov-Takens bifurcations in anapplication

P. De Maesschalck1, M. Wechselberger2

We study a more degenerate version of the well-known slow-fast Van der Polsystem, this time with a singularity that singularly unfolds as a Bogdanov-Takensbifurcation. We base ourselves on the local study performed in [1], complement itwith a global geometric singular perturbation analysis (in [2]) to give a thoroughview of all involved bifurcations such as Hopf, Homoclinic, SNIC bifurcations.

References

[1] De Maesschalck, P. and Dumortier, F., Slow-fast Bogdanov-Takens bifur-cations, Journal of Differential Equations 250 (2) (2011), 1000–1025.

[2] De Maesschalck, P. and Wechselberger, M., Neural excitability and sin-gular bifurcations, Journal of Mathematical Neuroscience 5 (16) (2015), 1–32.

1Department of Mathematics and Statistics, Hasselt University, Agoralaan,gebouw D, 3590-Diepenbeek, [email protected]

2School of Mathematics and Statistics, University of Sydney, NSW 2006, [email protected]

70



The role of chaos in the formation of binary objectsin the Kuiper-belt

David Farrelly1

Kuiper-belt binaries (KBBs) provide an invaluable window into conditions in theprimordial solar system. Several mechanisms for the formation of KBBs have beenproposed including; two-body collisions inside the Hill sphere of a larger body (Wei-denschilling, Icarus, 160, 212, 2002); strong dynamical friction (Goldreich, et al., Na-ture, 420, 643, 2002) and exchange reactions (Funato, et al., Nature, 427, 518, 2004).We propose a model of Kuiper-belt binary formation in the Hill approximation; themechanism involves the following sequence of events: (i) long-lived quasi-bound bi-naries are formed when two large objects (∼ 100 km sized) penetrate their mutualHill sphere and get caught up in a sea of chaos; (ii) the binary is stabilized throughgravitational scattering with a small intruder; and, finally, (iii) subsequent intruderscattering events gradually reduce the size of the binary until the mutual orbit isessentially Keplerian. In agreement with observations, the model predicts a propen-sity for the production of large mass ratio binaries having large semimajor axes andmoderately eccentric mutual orbits.

1Utah State UniversityLogan, UT, [email protected]

71



Summability of canard-heteroclinic saddleconnections

Karel Kenens1

For a given (real analytic) slow-fast system

x = εf(x, y, ε)

y = g(x, y, ε),

that admits a slow-fast saddle and that satisfies some mild assumptions, the Gevrey-summability properties of the saddle separatrix tangent in the direction of the criticalcurve are investigated: 1-summability is shown. It is also shown that slow-fast saddleconnections of canard type have summability properties, in contrast to the typicallack of Gevrey-summability for canard solutions of general equations.

1Department of Mathematics, Hasselt University, Martelarenlaan 42, 3500Hasselt, [email protected]

72



Exterior discrete semi-flows and applications

Miguel Maranon Grandes1

An exterior discrete semi-flow is a discrete semi-flow generated by an exteriorcontinuous map. The aim of this talk is to explain the notion of exterior discrete semi-flow and to apply it to the analysis of iterative processes induced by numerical methods–in this way, we are connecting some novel working purely topological techniques toseveral notions regarding dynamical systems. In particular, we have developed newcomputer programs for visualizing basins of attraction associated with the end pointsof a discrete semi-flow induced by a rational function defined on S2 and for computingthe measure of these basins, up to a certain precision.

1Department of Mathematics and Computer Science, University of La Rioja,Edificio J. L. Vives, C/ Luis de Ulloa s/n, 26004 Logrono, La Rioja, [email protected]

73



Dissipativity in nonautonomous linear-quadraticcontrol processes

Carmen Nunez1

This talk concerns the concept of the Willems dissipativity for nonautonomouslinear-quadratic control problems. A nonautonomous system of Hamiltonian ODEscan be associated with such a linear-quadratic problem, and the analysis of the cor-responding symplectic dynamics provides valuable information on the dissipativityproperties. The existence of globally defined nonnegative solutions of the Riccatiequation provided by the Hamiltonian system ensures the dissipativity and providesthe way to define the optimal storage function. The existence of such solutions isanalyzed in the scenarios of exponential dichotomy and weak disconjugacy, playingalso with the occurrence or absence of null controllability properties.

The work is made in collaboration with Roberta Fabbri, Russell Johsnon, SylviaNovo and Rafael Obaya.

1Departmento de Matematica Aplicada, Universidad de Valladolid, EII - Paseodel Cauce 79 - 47011 Valladolid - [email protected]

74



The parabolic RTBP. Interchange of mass after aclose encounter between galaxies

M. Olle1, E. Barrabes2, J.M. Cors3 L. Garcia4

We consider the motion of the Parabolic restricted three body problem (PRTBP).The goal of this problem is to study the motion of a massless body attracted, underthe Newton’s law of gravitation, by two masses moving in parabolic orbits all overin the same plane. The PRTBP may be regarded as a simplified model for themotion of two galaxies, taken as the primaries, and an infinitessimal mass. In orderto discuss possible motions for the particle, first we consider a rotating and pulsatingframe where the primaries remain at rest. For the system of ODE obtained we applydynamical systems tools. More precisely, this system of ODE is gradient-like and hasexactly ten hyperbolic equilibrium points lying on the boundary invariant manifoldscorresponding to escape of the primaries in past and future time. The invariantmanifolds of the equilibrium points play a key role in the dynamics and we studysome trajectories described by the particle before and after a close encounter betweenthe primaries. Finally some numerical simulations are done, paying special attentionto capture and escape orbits.

1Departament de Matematiques, Universitat Politecnica de Catalunya, Diag-onal 647, 08028 Barcelona, [email protected]

2,4Department d’Informatica, Matematica aplicada i Estadıstica, Universitat deGirona, Campus Montilivi, 17071 Girona, [email protected], [email protected]

3Departament de Matematiques, Universitat Politecnica de Catalunya, Av.Bases de Manresa 61-73, 08242 Manresa, [email protected]

75



Persistence of non-autonomous Nicholson’s systems

Ana M. Sanz1

Since in 1980 Gurney et al. [1] presented the scalar delay equation x′(t) = −µx(t)+p x(t−τ) e−γ x(t−τ) , which was called the Nicholson’s blowflies equation, many authorshave been concerned with the stability, persistence or existence of certain kind ofsolutions for this equation or for some of its generalizations. More recently Nicholsonsystems have also been considered, as they fit models for one single species in anenvironment with a patchy structure or for multiple biological species. For the caseof non-autonomous Nicholson systems, with a certain recurrent variation of the timeso that the theory of non-autonomous dynamical systems can be applied, we givenecessary and sufficient conditions for the presence of uniform persistence and strictpersistence. This talk is based on joint works with Sylvia Novo and Rafael Obaya.

References

[1] W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholson’s blowflies revisited,Nature 287 (1980), 17–21.

[2] S. Novo, R. Obaya, A.M. Sanz, Uniform persistence and upper Lyapunovexponents for monotone skew-product semiflows. Nonlinearity 26 (2013), 1–32.

[3] R. Obaya, A.M. Sanz, Uniform and strict persistence in monotone skew-productsemiflows with applications to non-autonomous Nicholson systems. Submitted forpublication (2016).

1Departamento de Didactica de las Ciencias Experimentales, Sociales y de laMatematica, and member of IMUVAUniversidad de ValladolidAvenida de Madrid 44, 34004 Palencia, [email protected]

76



Expanding Baker Maps: A class of piecewise linear2-D maps

Enrique Vigil Alvarez(in collaboration with A. Pumarino and J. A. Rodrıguez (UOV))

The main aim in [1], [2] and [3] is the study of certain two-dimensional maps thathave been called Expanding Baker Maps (EBMs for short). These maps, as the nametries to explain, reproduce the method used by a baker to knead the dough sincebakers bend and stretch a (quasi) two-dimensional domain over and over again untila final product is obtained: fairy cakes, bread rolls, country bread... In our terms,the final product is the attractor arising in the corresponding dynamics.

Although EBMs are, simply, piecewise linear maps defined in R2, they usuallydisplay very intricate and interesting dynamics. In this talk the concept of EBMwill be introduced. Moreover, we will show how these EBMs arise, the first resultsobtained for EBMs and certain approaches for future work in this field.

References

[1] J. F. Alves, A. Pumarino and E. Vigil: Statistical stability for multidimensionalpiecewise expanding maps. (sometido para publicacion) (2015)

[2] A. Pumarino, J. A. Rodrıguez, J. C. Tatjer and E. Vigil, Chaotic dynamics for2-d tent maps. Nonlinearity, 28, 407–434 (2015).

[3] E. Vigil, Chaotic dynamics for 2-d tent maps. (Doctoral Dissertation) Universityof Oviedo (2015).

1CMUP, University of PortoRua do Campo Alegre, 687 4169-007 [email protected]

77



Normal forms near a symmetric planar saddleconnection

Jeroen Wynen1

We consider vector fields of the formx =

(q2 + O(1 − x2)

)(1 − x2) + O(y),

y =(px + O(1 − x2)

)y + O(y2),

which contain a separatrix connection between hyperbolic saddles with opposite eigen-values where the connection is fixed. These situations appear in the local study ofnon-elementary singular points or after compactification of the phase space. We pro-vide smooth semi-local normal forms in vicinity of the connection, both in the resonantand non-resonant case. First, a formal conjugacy is constructed near the separatrix.Then, a smooth change of coordinates is realized by generalizing known local resultsnear the hyperbolic points.

1Department of Mathematics, Hasselt University, Martelarenlaan 42, 3500Hasselt, [email protected]

78



Spatio-temporal P-splines models in Bayesiandisease mapping

A. Adin1,2, M.D. Ugarte1,2, T. Goicoa1,2

In recent years, models incorporating splines have been considered for smoothingmortality or incidence risks in spatio-temporal disease mapping as an alternativeto conditional autoregressive (CAR) models. These models have been commonlyformulated within a hierarchical Bayesian framework with two main approaches: anEmpirical Bayes (EB) and a fully Bayes (FB) approach. The reformulation of the P-splines as generalized linear mixed models (GLMM) has been commonly consideredwithin the EB approach [1, 2], using the well-known penalized quasi-likelihood (PQL)technique for model fitting and inference. From a fully Bayes approach, Markov chainMonte Carlo (McMC) methods have been used to compute the posterior marginaldistributions of the splines’s regression coefficients [3]. Although these models arevery flexible, they can be computationally demanding to analyze spatio-temporaldata.

In this work, several models including one, two and three-dimensional P-splinesare proposed for smoothing risks in spatio-temporal disease mapping, fitting the mod-els from a fully Bayesian approach using integrated nested Laplace approximations(INLA). Specifically, spatially structured one-dimensional temporal P-splines, as wellas temporally structured two-dimensional spatial P-splines are proposed as an al-ternative to three-dimensional P-splines. Two real cancer data sets will be used toillustrate these models.

References

[1] M.D. Ugarte, T. Goicoa, and A.F. Militino, Spatio-temporal modeling ofmortality risks using penalized splines, Environmetrics, 21 (2010), 270 – 289.

[2] M.D. Ugarte, T. Goicoa, J. Etxeberria, and A.F. Militino, A P-splineANOVA type model in space-time disease mapping, Stochastic EnvironmentalResearch and Risk Assessment, 26 (2012), 835 – 845.

[3] Y.C. MacNab, and P. Gustafson, Regression B-spline smoothing in Bayesiandisease mapping: with an application to patient safety surveillance, Statistics inMedicine, 26, (2007), 4455 – 4474.

1Department of Statistics and O.R., Public University of Navarre, Campus deArrosadıa, 31006 Pamplona, Spain.

2Institute for Advanced Materials (InaMat), Public University of Navarre,[email protected], [email protected], [email protected]

79



Recent developments in Stein’s method

Christian Dobler1

Stein’s method of distributional approximation has become a popular techniquefor estimating the distance between the distribution of a given random variable Wand that of another random variable Z whose law can be characterized by some lineardifferential or difference operator. Being first established for the univariate normal dis-tribution by C. Stein it has been extended over the years to many other distributionslike the Poisson, gamma, exponential, beta and multivariate normal distributions byvarious authors. In this talk we present recent results on the normal approximationof non-linear functionals of independent random variables like degenerate U -statistics[1] and random sums [2] which have been derived via Stein’s method. If time allows,then we will also discuss the case of Gamma approximation of degenerate U -statisticsand/or the development of Stein’s method for functions of multivariate normal vectorsfrom [3] and [4] as well as suitable applications.

References

[1] C. Dobler and G. Peccati, Quantitative de Jong theorems in any dimension,arXiv:1603.00804 (2016).

[2] C. Dobler, New Berry-Esseen and Wasserstein bounds in the CLT for non-randomly centered random sums by probabilistic methods, ALEA - Latin Amer-ican Journal of Probability and Mathematical Statistics XII, no. 2 (2015), 863 –902.

[3] R. E. Gaunt, Stein’s method for functions of multivariate normal random vari-ables, arXiv:1507.08688 (2015).

[4] R. E. Gaunt and G. Reinert, The rate of convergence of some asymptoticallychi-square distributed statistics by Stein’s method, arXiv:1603.01889v2 (2016).

1Mathematics Research Unit, University of Luxembourg,Campus Kirchberg, Universite du Luxembourg, 6, rue Richard Coudenhove-Kalergi L-1359 [email protected]

80



Nonparametric conditional copula estimation andapplications

I. Gijbels

A convenient way to model dependency is through a copula function, that cou-ples the (conditional) joint distribution with the (conditional) marginal distributions.In recent years a lot of attention has been devoted to statistical inference for con-ditional copulas and resulting conditional association measures. See, for example,[1] and [2]. In dependence modelling using conditional copulas, one often imposesthe working assumption that the covariate(s) influences the conditional copula solelythrough the marginal distributions. This so-called (pairwise) simplifying assumptionis almost standardly made in vine copula constructions. When such a simplifyingassumption holds, this knowledge should lead to a more efficient estimation of thedependence structure. In this talk we briefly discuss nonparametric estimation of aconditional copula in a general setting and in the simplifying assumption (cf. [3]). Asexemplified in the literature, such an assumption might not be justified (see e.g. [4]).We briefly discuss how one could proceed to test whether this assumption holds or not.

This talk is based on joint work with M. Omelka and N. Veraverbeke.

References

[1] I. Gijbels, N. Veraverbeke, and M. Omelka, Conditional copulas, associa-tion measures and their applications, Computational Statistics & Data Analysis55 (2011), 1919–1932.

[2] N. Veraverbeke, M. Omelka and I. Gijbels, Estimation of a conditional cop-ula and association measures, The Scandinavian Journal of Statistics 38 (2011),766–780.

[3] I. Gijbels, M. Omelka and N. Veraverbeke, Estimation of a copula whena covariate affects only marginal distributions, Scandinavian Journal of Statistics42 (2015), 1109–1126.

[4] E.F. Acar, C. Genest, J. Neslehova, Beyond simplified pair-copula construc-tions, Journal of Multivariate Analysis 110 (2012), 74-90.

Department of Mathematics and Leuven Statistics Research Center

KU Leuven, Box 2400, Celestijnenlaan 200B, B-3001 Leuven (Heverlee), Belgium

[email protected]

81



Targeted penalized regression for cost effectivecausal effect estimation

Machteld Varewyck1, Stijn Vansteelandt1, Marie Eriksson2 ElsGoetghebeur1

In the absence of unmeasured confounders, standardised risks derived from con-founder adjusted regression of outcome on exposure can form the basis of causal effectestimators. The method is in principle simple and beats the more sophisticated dou-ble robust estimators in specific settings. Both can be applied for the measurement ofquality of care, e.g. to evaluate the added effect of hospital choice on patient survival.In practice, even the simple regression based measure meets substantial statisticaland numerical challenges (cf. [2]) when a large number of centres is evaluated in thesingle model. Because of overfitting the resulting estimator can face convergence andstability problems, and it may suffer substantial finite sample bias and large vari-ance. Normal random effects models have been applied to overcome this, but loosesubstantial power for the detection of poor performance in small centres. We showhow Firth’s correction (cf. [1]), originally developed as an asymptotic first order biascorrection for maximum likelihood estimators, strikes a balance between fixed andnormal random effects regression that gives enough prior weight to outlying risks torecover much needed power (cf. [3]). Moving on from this we develop an adaptedlasso penalisation for model selection that accounts for the cost of measuring specificcovariates. In our setting it involves a cross validation method that targets causal ef-fects which are not directly observed. We develop and apply the approach to estimatequality of care based on the Swedish RiksStroke register.

References

[1] D. Firth, Bias reduction of maximum likelihood estimates. Biometrika 80 (1993)27-38.

[2] D. Fouskakis and D. Draper, Comparing stochastic optimization methods forvariable selection in binary outcome prediction, with application to health policy.Journal of the American Statistical Association 103 (2008) 1367–1381.

[3] M. Varewyck, E. Goetghebeur, M. Eriksson and S. Vansteelandt, Onshrinkage and model extrapolation in the evaluation of clinical center performance.Biostatistics 15 (2014) 651–664.

1Department of Applied Mathematics, Computer Science and Statistics,Ghent University, Krijgslaan 281-S9, 9000 Ghent,[email protected]

2Department of Statistics, Umea University,[email protected]

82



A continuous updating weighted least squaresestimator of tail dependence in high dimensions

John H.J. Einmahl1, Anna Kiriliouk2, Johan Segers2

Likelihood-based procedures are a common way to estimate tail dependence pa-rameters. They are not applicable, however, in non-differentiable models such as thosearising from recent max-linear structural equation models. Moreover, they can behard to compute in higher dimensions. An adaptive weighted least-squares procedurematching nonparametric estimates of the stable tail dependence function with thecorresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range ofsampling schemes and tail dependence models. In large samples, it is asymptoticallynormal with an explicit and estimable covariance matrix. The minimum distanceobtained forms the basis of a goodness-of-fit statistic whose asymptotic distributionis chi-square. The estimator is applied to disentangle sources of tail dependence inEuropean stock markets.

1Department of Econometrics & OR and CentERTilburg UniversityP.O. Box 90153, 5000 LE Tilburg, the [email protected]

2Institute de Statistique, Biostatistique et Sciences ActuariellesUniversite catholique de LouvainVoie du Roman Pays 20, B-1348 Louvain-la-Neuve, [email protected], [email protected]

83



SAS distributions

Arthur Pewsey1

Sinh-arcsinh (SAS) distributions were first proposed in [1] and arise from the ap-plication of the SAS transform to a continuous random variable that is symmetricallydistributed about the origin. In addition to location and scale parameters, SAS distri-butions have two extra parameters controlling skewness and kurtosis. Consequently,the forms they can adopt are extremely wide-ranging. The special cases of the sinh-arcsinhed normal (SASN), t (SAST) and logistic (SASL) classes have been studied in[1], [5] and [4], respectively, and employed to model stochastic scenarios as diverse ascrude oil production (cf. [2]), central limit theorems under special relativity (cf. [3])and gamma-ray burst duration (cf. [6]). SASN densities are always unimodal, whereasSAST and SASL densities can be uni- or bimodal. I will summarize the basic prop-erties of SAS distributions, many of which are highly appealing and some of whichare perhaps surprising, and present new results for quantile-based estimation of theirparameters and the performance of edf-based parametric bootstrap goodness-of-fittests.

References

[1] M. C. Jones and A. Pewsey, Sinh-arcsinh distributions, Biometrika 96 (2009),761–780.

[2] K. Matsumoto, V. Voudouris, D. Stasinopoulos, R. Rigby and C. DiMaio, Exploring crude oil production and export capacity of the OPEC MiddleEast countries, Energy Policy 48 (2012), 820–828.

[3] I. W. McKeague, Central limit theorems under special relativity, Statistics andProbability Letters 99 (2015), 149–155.

[4] A. Pewsey and T. Abe, The sinh-arcsinhed logistic family of distributions:properties and inference, Annals of the Institute of Statistical Mathematics 67(2015), 573–594.

[5] J. F. Rosco, M. C. Jones and A. Pewsey, Skew t distributions via the sinh-arcsinh transformation, Test 20 (2011), 630–652.

[6] M. Tarnopolski, Analysis of gamma-ray burst duration distribution using mix-tures of skewed distributions, Monthly Notices of the Royal Astronomical SocietyDOI:10.1093/mnras/stw429 (2016).

1Department of Mathematics, Escuela Politecnica, University of Extremadura,Avenida de la Universidad s/n, 10003 Caceres, [email protected]

84



Non-Universality of Nodal Length Distribution forArithmetic Random Waves

Maurizia Rossi1

“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus. We are interested in the distribution of the length of their nodallines. In [1] the authors prove that the asymptotics for the variance is non-universal.Their result is intimately related to the arithmetic of lattice points lying on a circlewith radius corresponding to the energy.

In this talk we show that the nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice pointslying on circles. Our argument has two main ingredients. An explicit derivation ofthe Wiener-Ito chaos expansion for the nodal length shows that it is dominated by its4th order chaos component (in particular, somewhat surprisingly, the second orderchaos component vanishes - this is closely related to the so-called “obscure” Berry’scancellation phenomenon). The rest of the argument relies on the precise analysis ofthe fourth order chaotic component.

This talk is based on [2], joint work with Domenico Marinucci (Universita diRoma Tor Vergata), Giovanni Peccati (Universite du Luxembourg) and Igor Wigman(King’s College London).

References

[1] M. Krishnapur, P. Kurlberg, and I. Wigman, Nodal length fluctuations forarithmetic random waves, Annals of Mathematics 177 (2013), 699–737.

[2] D. Marinucci, G. Peccati, M. Rossi, and I. Wigman, Non-universalityof nodal length distribution for arithmetic random waves, Preprint, arXiv1508.00353 (2015).

1Unite de Recherche en Mathematiques, Universite du Luxembourg, 6, rueCoudenhove-Kalergi, L-1359 [email protected]

85



Circular Isotonic Regression with Applications tocell-cycle Biology

Cristina Rueda1, Miguel Fernandez2

Constraints on parameters arise naturally in many applications. Statistical isotonicregression methods that honor the underlying constraints tend to be more powerfuland result in better interpretation of the underlying scientific data.

While Euclidean space data are commonly encountered in applications, there arenumerous instances where the underlying data and the parameters of interest resideon a unit circle that are often the result of an oscillatory system. One example ofan oscillatory systems is the periodic expression of genes participating in cell divisioncycle. In this case, researchers are interested in correlating the phases of periodicgenes across different experimental conditions or species or tissues etc. Thus thestatistical problem of interest is to draw inferences regarding the relative order amongparameters on a unit circle.

There exists a long history of statistical literature on isotonic regression in thecontext of Euclidean space data (cf. [3]), and also on methodology for analyzingangular data (cf. [2]). In comparison, isotonic inference for circular data is almostnon-existent. Just as one cannot trivially extend standard statistical methods in theEuclidean space to the circle, isotonic statistical regression for Euclidean space cannotbe extended to constraints on a unit circle (cf. [1]). The purpose of this talk is two-fold. First we describe recent theoretical and methodological advances in CircularIsotonic Regression and second we describe some applications of the methodology incell biology.

References

[1] C. Rueda, M.A. Fernandez, and S.Peddada, Estimation of Parameters Sub-ject to Order Restrictions on a Circle with Application to Estimation of PhaseAngles of Cell-Cycle Genes, Journal of the American Statistical Association 104,485 (2009), 338–347.

[2] K. Mardia and P. Jupp, Directional Statistics. John Wiley & Sons, New York,2000.

[3] M.J. Silvapulle and P.K. Sen, Constrained Statistical Inference. Inequality,Order and Shape Restrictions. John Wiley & Sons, New York, 2004.

1Department of Statistics and Operation Research, University of Valladolid,Paseo Belen 7, [email protected]

2Department of Statistics and Operation Research, University of Valladolid,Paseo Belen 7, [email protected]

86



Integrated likelihood based inference for nonlinearpanel data models with unobserved effects

Martin Schumann1, Thomas A. Severini2, Gautam Tripathi1

Panel data models are used widely by economists and other social scientists tocapture the effects of unobserved individual heterogeneity. In this paper, we proposea new general approach for estimating panel data models when the unobserved indi-vidual effects enter the model nonlinearly. The asymptotic statistical theory for theproposed estimator is developed in a setting where the number of individuals andthe number of time periods both approach infinity. Results from a simulation studysuggest that our methodology can work very well even in small to moderately sizedsamples.

1Center for Research in Economics and Management (CREA), University ofLuxembourg, L-1511, [email protected], [email protected]

2Department of Statistics, Northwestern University, Evanston, IL-60201,[email protected]

87



An applied approach to robust statistical analysisof the location of interval-valued data

Beatriz Sinova1,3 and Stefan Van Aelst2,3

The need for statistical analysis of interval-valued data arises with the increasingnumber of real-life experiments whose outputs are imprecise and require intervals formodelling such imprecision. This imprecision can result from different situations: datacan be essentially interval-valued, like interval-censored data, but they can also cor-respond to aggregate information because the magnitude of interest is the fluctuationof a real-valued attribute over a given time period or collection of individuals or dueto confidentiality reasons, or they can even originate as a consequence of uncertainor incomplete information. In order to illustrate the importance of interval-valueddata in knowledge fields as different as Finance, Chemistry or Social Sciences, somereal-life examples will serve as motivation.

The generalization of statistical techniques and procedures to cover the interval-valued setting is usually not straightforward because of the peculiarities of the consid-ered space. The space of non-empty compact intervals, Kc(R), is not linear with theusual interval arithmetic, but forms a closed convex cone. One of the consequencesis that there is no ‘difference operation’ that is always well-defined and preserves themain properties of the difference between real values in connection with the sum. Inpractice, such a drawback is overcome to some extent by replacing differences in thestatistical developments by suitable distances between values in Kc(R).

Most of the statistical techniques already adapted for interval-valued data arebased on the Aumann mean as location measure. Despite its numerous handy prop-erties, the Aumann mean is an extension of the concept of mean of a random variable,from which it inherits the high sensitivity to outliers and data changes, as will be il-lustrated through some real-life examples. Thus, a more robust alternative measureshould be used instead of the Aumann mean in situations involving data contami-nation, so frequently encountered in applications. The aim of this work will be toprovide robust tools to summarize the location of interval-valued data and to intro-duce them from an applied point of view, highlighting the advantages of their useand showing their suitable performance by means of the motivating real-life studies.Finally, some concluding remarks about future research lines related to this work willbe given.

1Department of Statistics and Operational Research and D. M.,University of Oviedo, 33007 Oviedo, [email protected]

2Department of Mathematics, KU Leuven, 3001 Leuven, [email protected]

3Department of Applied Mathematics, Computer Science and Statistics,Ghent University, 9000 Gent, Belgium

88



Stein’s method and the influence of the prior inBayesian statistics

Yvik Swan1

Joint work with Gesine Reinert (Oxford) and Christophe Ley (Ghent)

We propose tight upper and lower bounds for the Wasserstein distance between anytwo univariate continuous distributions with probability densities p1 and p2 havingnested supports. These explicit bounds are expressed in terms of the derivative of thelikelihood ratio p1/p2 as well as the Stein kernel of p1. The method of proof relies ona new variant of Stein?s method which manipulates Stein operators. We give severalapplications of these bounds. Our main application is in Bayesian statistics : wederive explicit data-driven bounds on the Wasserstein distance between the posteriordistribution based on a given prior and the no-prior posterior based uniquely on thesampling distribution. This is the first finite sample result confirming the well- knownfact that with well-identified parameters and large sample sizes, reasonable choices ofprior distributions will have only minor effects on posterior inferences if the data arebenign.

1Universite de [email protected]

89



Heat kernel formulae and theBrownian bridge to a submanifold

James Thompson1

Heat kernels are important for a variety of reasons. They are connected to thestudy of the heat equation, can be used as analytic tools and capture certain geometricand topological properties of the underlying space. For Riemannian manifolds with apole, the minimal heat kernel has a probabilistic formula due to K.D. Elworthy andA. Truman (cf. [1]). By a joint with X.-M. Li (cf. [2]), this can be adapted to alsoprovide formulae for the derivatives of the heat kernel.

In this talk, we present a more general formula, for the minimal heat kernel on anycomplete Riemannian manifold. This object coincides with the transition densities ofBrownian motion, whose construction on a manifold we will review. The Brownianbridge is given by conditioning Brownian motion to hit a fixed point at a fixed positivetime. We extend this concept by replacing the fixed point with a submanifold and useour formula to derive lower bounds, an asymptotic relation and derivative estimatesfor the conditional measure.

The motivation for this is the desire to extend the analysis of path and loop spacesto measures on paths which terminate on a submanifold, the intention being to studythe relationship between the geometry of the path space, the intrinsic geometry ofthe ambient manifold and the extrinsic geometry of the submanifold.

References

[1] K. D. Elworthy and A. Truman, The diffusion equation and classical mechan-ics: an elementary formula, Stochastic processes in quantum theory and statisticalphysics 173 (1982), 136–146.

[2] X.-M. Li and J. Thompson, First and second order Feynman-Kac formulae,Preprint (2016).

[3] J. Thompson, Brownian bridges to submanifolds, Preprint arXiv:1604.05182(2016).

1Mathematics Research Unit,University of Luxembourg,6, rue Coudenhove-Kalergi,L-1359 [email protected]

90



Weighted inequalities for the Riesz potential on thesphere

Alberto Arenas, Oscar Ciaurri, Edgar Labarga

The fractional integral operator (or Riesz potential) of the classical Laplacian isgiven by

Tσf(x) =

∫

Rd

f(y)

|x− y|d−σ dy, 0 < σ < d.

Stein and Weiss (cf. [2]) proved that, under suitable conditions on the parameters,the fractional integral operator satisfies the inequality

‖|x|−aTσf‖Lp(Rd) ≤ ‖|x|bf‖Lq(Rd).

This result cannot be improved for general functions in Lq(Rd). However, De Napoli,Drelichman, and Duran (cf. [1]) extended the range of admissible power weights withf a radial function.

In this talk, we prove a version of the Stein-Weiss inequality in the setting of thed-dimensional sphere Sd. In this space the role of the fractional integral Tσ is playedin the sphere by the operator

Aσf(η) =Γ(√−Lλ + 1−σ

2 )

Γ(√−Lλ + 1+σ

2 )f(η), η ∈ Sd−1,

where −Lλ is the conformal Laplacian on the sphere Lλ = −∆0 + λ2, λ = (d− 2)/2.The operator Aσ can be written in terms of the Riesz potential on the sphere, so it iscalled the Riesz potential of the conformal Laplacian. So we shall prove the inequality

‖(|η − ed||η + ed|)−afAσf‖Lp(Sd−1) ≤ C‖(|η − ed||η + ed|)b‖Lq(Sd−1).

We also improve the range of admissible power weights for functions which areinvariant under the action of the group SO(d − 1) on Sd−1, the analogous to radialfunctions on the sphere.

References

[1] P. L. De Napoli, I. Drelichman, and R. G. Duran, On weighted inequalitiesfor fractional integrals of radial functions, Illinois J. Math. 55 (2011), 575–587.

[2] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclideanspace, J. Math. Mech. 7 (1958), 503–514.

1Department of Mathematics and Computer Science, University of La Rioja,Calle Luis de Ulloa s/n, 26004 Logrono, Spainalarenas,oscar.ciaurri,[email protected]

91



Critical points of polynomials

Manuel Bello Hernandez1,

This talk is concerned with properties of the critical points of orthogonal polyno-mials with respect to a measure on the unit circle (OPUC). The main result statesthat the asymptotic distribution of the critical points of OPUC coincides with theasymptotic distribution of its zeros and Nevai-Totik points attract the same num-ber of critical points as zeros of the OPUC. Analogous results are also presented forpara-orthogonal polynomials and for orthogonal polynomials with respect to a regularmeasure supported on a continuum set.

References

[1] H.-P. Blatt, E. B. Saff, and M. Simkani, Jentzsch–Szego type theorems forthe zeros of best approximants, J. London Math. Soc. 38 (1988), 307–316.

[2] B. de la Calle Ysern, The Jentzsch–Szego Theorem and Balayage Measures,Const. Approx. 40 (2014), 307–327.

[3] J. Degot, Sendov conjecture for high degree polynomials, Proc. AMS 142 (2014),1337–1349.

[4] P. Nevai and V. Totik, Orthogonal polynomials and their zeros, Acta Sci.Math. (Szeged) 53 (1989), 99–104.

[5] B. Simon, Fine structure of the zeros of orthogonal polynomials. II. OPUC withcompeting exponential decay. J. Approx. Theory 135 (2005), 125–139.

1Departamento de Matematicas y Computacion, Universidad de La Rioja,Edif. J. L. Vives, Calle Luis de Ulloa, n. 2, 26004 Logrono, [email protected]

92



Darboux for CMV and Christoffel transformations

M.J. Cantero1, F. Marcellan2 L. Moral1 L. Velazquez1

Orthogonal polynomials on the real line satisfy a recurrence relation which can beexpressed in terms of Jacobi matrices, canonical representations of self-adjoint oper-ators. The Darboux transformations for Jacobi matrix are equivalent to Christoffelmodifications of the corresponding orthogonality measures. This has deep implica-tions in the study of some integrable systems.

The unitary analogue of Jacobi matrices is given by CMV matrices, canonicalrepresentation of unitary operators, and closely related to (Laurent) orthogonal poly-nomials on the unit circle (OPUC).

Despite the multiple connections between OPUC and other fields (among them,integrable systems) opened by CMV matrices, there is no Darboux transformationfor CMV yet.

We will show how to define Darboux transformation for CMV matrices by analogywith the Jacobi case. This analogy is highlighted even more by the connection betweenDarboux for CMV and Christofell transformation of measures on the unit circle. Asin the Jacobi case, Darboux for CMV not only connect CMV matrices, but also therelated OPUC.

1Department of Applied MathematicsUniversity of ZaragozaPedro Cerbuna, 1250009, [email protected], [email protected], [email protected]

2Department of MathematicsUniversity Carlos III of MadridAvenida de la Universidad,3028911, Leganes, [email protected]

93



Positive quadrature formulas on the unit circle withprescribed nodes. A new approach

Ruyman Cruz-Barroso1,

In this talk we consider the approximation of integrals with respect to measuressupported on [−π, π] by means of positive quadrature formulas on the unit circle withmaximal domain of exactness and possibly some preassigned nodes. The quadratureformulas are exact in a new nested sequence of subspaces of Laurent polynomials thatbrings better properties to the rules, collaborating thus to formalize an own theorythat allow us to obtain analogous properties to the usual quadrature formulas onthe real line. We guarantee the existence of two positive rules of maximal domainof exactness, and we characterize the existence of positive quadrature formulas thatshould have up to three nodes prescribed in advance. We obtain estimations of theerror for these quadrature formulas, depending on the regularity of the integrand.

The content of this talk is a part of a joint work in progress in collaboration withCarlos Dıaz Mendoza and Francisco Perdomo-Pıo.

1Department of Mathematical Analysis,La Laguna University,La Laguna, Santa Cruz de Tenerife, 38271, Canary Islands, Spain.

[email protected]

94



A higher rank generalization of the Bannai-Itoalgebra

Hendrik De Bie1, Vincent X. Genest2, Luc Vinet3

The kernel of the Zn2 Dirac-Dunkl operator is examined by determining the symme-

try algebra An of the associated Dirac-Dunkl equation, which is shown to correspondto a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomialnull-solutions of the Dirac-Dunkl operator is constructed. The basis elements are jointeigenfunctions of a maximal commutative subalgebra of An and are given explicitlyin terms of Jacobi polynomials. The symmetry algebra is shown to act irreduciblyon this basis via raising/lowering operators. A scalar realization of An is proposed.Finally, it is conjectured that the multivariate Bannai-Ito polynomials will arise asoverlap coefficients between different sets of basis elements.

References

[1] H. De Bie, V. X. Genest, L. Vinet, A Dirac-Dunkl equation on S2 and theBannai-Ito algebra. To appear in Comm. Math. Phys.

[2] H. De Bie, V. X. Genest, L. Vinet, The Zn2 Dirac–Dunkl operator and a

higher rank Bannai–Ito algebra. arXiv:1511.02177.

1Department of Mathematical Analysis, Faculty of Engineering and Architec-ture, Ghent University Galglaan 2, 9000 Gent, [email protected]

2Department of Mathematics, Massachusetts Institute of Technology, 77 Mas-sachusetts Ave, Cambridge, MA 02139, [email protected]

3Centre de Recherches Mathematiques, Universite de Montreal, P.O. Box6128, Centre-ville Station, Montreal, QC H3C 3J7, [email protected]

95



Orthogonal polynomials, partition functions andasymptotics

Alfredo Deano1

The asymptotic analysis of orthogonal polynomials, on the real line as well as inthe complex plane, has proven very fruitful in the study of the partition functionand free energy of certain ensembles of random matrices, as the size of the matricesN tends to infinity. In this talk, we will present the general theory and illustrate itwith two examples, both with weights of the form w(z) = e−NV (z), where V (z) is apolynomial of degree 2 or 3.

This is joint work with Pavel M. Bleher and Maxim Yattselev (Indiana University-Purdue University Indianapolis, United States).

1School of Mathematics, Statistics and Actuarial Science,University of KentCanterbury CT2 7NFUnited [email protected]

96



Asymptotics of Sobolev Orthogonal polynomials onthe unit ball

A. M. Delgado1, L. Fernandez1, D. S. Lubinsky2, T. E. Perez1, M. A.Pinar1

Sobolev orthogonal polynomials on the unit ball are studied The correspondingSobolev inner product is defined involving outward normal derivatives on the sphere.We will give explicit representation for orthogonal polynomials and reproducing ker-nels in term of classical polynomials on the ball. From these explicit expressions, al-gebraic properties and asymptotic behaviour of Christoffel functions will be deduced.These results can be found in [1].

References

[1] A. M. Delgado, L. Fernandez, D. S. Lubinsky, T. E. Prez, andM. A. Piar, Sobolev orthogonal polynomials on the unit ball via outwardnormal derivatives, J. Math. Anal. Appl., Available online 18 March 2016,http://dx.doi.org/10.1016/j.jmaa.2016.03.041.

[2] C. F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, second edi-tion, Encyclopedia of Mathematics and Its Applications, Cambridge UniversityPress, Cambridge, 2014.

1Department of Applied Mathematics and IEMath–Math Institute, Universityof Granada, 18071 Granada, [email protected], [email protected], [email protected], [email protected]

2School of Mathematics, Georgia Institute of Technology, Atlanta, GA [email protected]

97



An analogue of Krall orthogonal polynomials on thesimplex

Lidia Fernandez

With a similar construction as the one done by Koornwinder in 1975, we obtain amutually orthogonal basis with respect to a Krall inner product on the simplex. Fur-thermore, using properties of the polynomials in one variable we get some interestingproperties of these polynomials on the simplex.

This is a joint work with Antonia M. Delgado, Teresa E. Perez and Miguel A.Pinar.

References

[1] C. F. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopediaof Mathematics and its Applications 81, Cambridge University Press, 2001.

[2] T. H. Koornwinder, Two variable analogues of the classical orthogonal poly-nomials, emphTheory and Application of Special Functions, R. Askey Editor,Academic Press (1975), 435–495.

Departamento de Matematica Aplicada, Universidad de Granada, Granada,[email protected]

98



Christoffel transformations for matrix orthogonalpolynomials in the real line

Carlos Alvarez-Fernandez 1, Gerardo Ariznabarreta2, Juan CarlosGarcıa-Ardila3 Manuel Manas2 Francisco Marcellan3

Given a matrix polynomial W (x), matrix bi-orthogonal polynomials with respectto the sesquilinear form

< P (x), Q(x) >W =

∫P (x)W (x)dµ(x)(Q(x))>, P,Q ∈ Rp×p[x],

where µ(x) is a matrix of Borel measures supported in some infinite subset of thereal line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to < ·, · >W and matrix polynomials orthogonalwith respect to µ(x) are presented. In particular, for the case of nonsingular leadingcoefficients of the perturbation matrix polynomial W (x) we present a generalizationof the Christoffel formula constructed in terms of the Jordan chains of W (x). Forperturbations with a singular leading coefficient several examples by Duran et al arerevisited.

References

[1] C. Alvarez-Fernandez, G. Ariznabarreta, J. C. Garcıa-Ardila, M.Manas, F. Marcellan, Christoffel transformations for matrix orthogonal poly-nomials in the real line and the non-Abelian 2D Toda lattice hierarchy, Internat.Math. Res. Notices, in press. arXiv:1511.04771v2 [math.CA].

1Departamento de Metodos Cuantitativos, Universidad Pontificia Comillas,Calle de Alberto Aguilera 23, 28015-Madrid, [email protected]

2Departamento de Fısica Teorica II (Metodos Matematicos de la Fısica), Uni-versidad Complutense de Madrid, Ciudad Universitaria, Plaza de Ciencias 1,28040-Madrid, [email protected], [email protected]

3Departamento de Matematicas, Universidad Carlos III de Madrid, AvenidaUniversidad 30, 28911 Leganes, [email protected], [email protected]

99



A Hardy inequality for ultraspherical expansionswith an application to the sphere

Alberto Arenas, Oscar Ciaurri, Edgar Labarga

The Hardy inequality for fractional powers of the classical Laplacian is

Cσ,d

∫

Rd

u2(x)

|x|2σ dx ≤∫

Rdu(x)(−∆u(x))σ dx,

where the sharp constant is well-known (cf. [1, 2]).Consider the family cλnn≥0 of orthonormal ultraspherical polynomials cλn on

L2((−1, 1), dµλ), dµλ(x) = (1−x2)λ−1/2 dx. Those polynomials are eigenfunctions of

the operator Lλ = (1− x2) d2

dx2 − (2λ+ 1)x ddx − λ2; that is, Lλcλn = −(n+ λ)2cλn.

Now define the operator

Aλσ =Γ(√−Lλ + 1+σ

2

)

Γ(√−Lλ + 1−σ

2

) .

In this talk we present a Hardy inequality for the operator Aλσ of the form

Qσ,λ

∫ 1

−1

u2(x)

(1− x2)σ/2dµλ(x) ≤

∫ 1

−1

u(x)Aλσu(x) dµλ(x),

where the constant Qσ,λ = 2σ Γ(λ/2+(1+σ)/4)2

Γ(λ/2+(1−σ)/4)2 is sharp.

As a result of this inequality we obtain a Hardy inequality on the sphere involveda potential having singularities in both poles of the sphere that takes the form

2σQσ,(d−1)/2

∫

Sd

f2(ξ)

(|ξ − ed||ξ + ed|)σdξ ≤

∫

Sdf(ξ)Aσf(ξ) dξ,

with Aσ =Γ(√

−∆Sd+ 1+σ2 )

Γ(√

−∆Sd+ 1−σ2 )

and −∆Sd being the conformal Laplacian on the sphere.

References

[1] W. Beckner, Pitt’s inequality and the fractional Laplacian: sharp error esti-mates, Forum Math. 24 (2012), 177–209.

[2] D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal.168 (1999), 121–144.

1Departamento de Matematicas y Computacion, Universidad de La Rioja,Calle Luis de Ulloa s/n, 26004 Logrono, Spainalarenas,oscar.ciaurri,[email protected]

100



Dynamics and interpretation of some integrablesystems via matrix orthogonal polynomials

Ana Mendes1, Amılcar Branquinho2, Ana Foulquie Moreno3

In this work we characterize a Toda type lattice in terms of a family of matrix poly-nomials orthogonal with respect to a complex matrix measure. In order to study thesolution of this dynamical system we give explicit expressions for the Weyl function,generalized Markov function, and we also obtain, under some conditions, a represen-tation of the vector of linear functionals associated with this system. We also provea Lax type theorem for the point spectrum of the Jacobi operator associated with aToda type lattice.

References

[1] O. I. Bogoyavlenskii, Some Constructions of integrable dynamical systems, Math.USSR Izv. 1988; 31 no. 1:47–75.

[2] O. I. Bogoyavlenskii, Integrable Dynamical Systems Associated with the KdVEquation, Math. USSR Izv. 1988; 31 no. 3:435–454.

[3] A. Aptekarev, V. Kaliaguine and J. V. Iseghem, Genetic sum’s representation forthe moments of a system of Stieltjes functions and its application, Constr. Approx.2000; 16:487-524.

[4] V. Sorokin and J. V. Iseghem, Matrix HermitePade problem and dynamical sys-tems, J. Comput. Appl. Math. 2000; 122:275–295.

[5] X. Geng, F. Li, and B. Xue, A generalization of Toda Lattices and their Bi-Hamiltonian structures, Modern Physics Letters B. 2012; 26 no. 13:1250078-1–1250078-7.

[6] D. Barrios, A. Branquinho, and A. Foulquie Moreno, On the full Kostant Todasystem and the discrete Korteweg-de Vries equations, J. of Math. Anal. and Appl.2013; 401 no.2:811–820.

[7] D. Barrios, A. Branquinho, and A. Foulquie Moreno, On the relation between thefull Kostant-Toda lattice and multiple orthogonal polynomials, J. of Math. Anal.and Appl. 2011; 377 no.1:228–238.

[8] D. Barrios, A. Branquinho, and A. Foulquie Moreno, Dynamics and interpretationof some integrable systems via multiple orthogonal polynomials, J. of Math. Anal.and Appl. 2010; 361 no.2:358–370.

[9] B. Beckermann, On the convergence of bounded J-fractions on the resolvent setof the corresponding second order difference operator, J. Approx. Theory. 1999;99 no. 2:369–408.

101

[10] Yu. M. Berezanskii, Integration of nonlinear difference equations by the inversespectral problem method, Doklady Russ. Acad. Nauk, 1985; 281 no.1, Engl. transl.in Soviet Math. Doklady. 1985; 31 no.2:264–267.

[11] A. Aptekarev, A. Branquinho, and F. Marcellan, Toda-Type differential equa-tions for the recurrence coefficients of orthogonal polynomials and Freud transfor-mation, J. Comput. Appl. Math., 1997; 78 no.1:139–160.

[12] L. Miranian, Matrix-valued orthogonal polynomials on the real line: some exten-sions of the classical theory, J. Phys. A: Math. Gen. 2005; 38:5731–5749.

[13] M. J. Cantero, L. Moral and L. Velazquez, Matrix orthogonal polynomials whosederivatives are also orthogonal, J. Approx. Theory, 2007; 74:174–211.

[14] A. J. Duran and M. Ismail, Differential coefficients of orthogonal matrix polyno-mials, J. Comput. Appl. Math., 2006; 190:424–436.

[15] A. J. Duran and F. A. Grunbaum, Orthogonal matrix polynomials, scalar-typeRodrigues formulas and Pearson equations, J. Approx. Theory, 2005; 134:267–280.

[16] P. D. Lax, Linear algebra and its applications, second edn. Pure and AppliedMathematics (Hoboken). Wiley-Interscience [John Wiley & Sons], Hoboken, NJ(2007).

[17] A. Branquinho, F. Marcellan, and A. Mendes, Vector interpretation of the matrixorthogonality on the real line, Acta Applicandae Mathematicae, 2010; 112 no.3:357–383.

[18] A. Branquinho, L. Cotrim and A. Foulquie Moreno, Matrix interpretation ofmultiple orthogonality, Numerical Algorithms, 2010; 55 no. 1:19–37.

[19] C. Berg, The Matrix Moment Problem, Coimbra Lecture Notes on OrthogonalPolynomials, A. Branquinho and A. Foulquie Moreno, Ed. Nova Publishers, NewYork 2008:1–57.

[20] A. J. Duran, A generalization of Favard’s Theorem for polynomials satisfying arecurrence relation, J. Approx. Theory, 1993; 74:83–109.

[21] A. J. Duran, Markov theorem for orthogonal matrix polynomials, Canad. J.Math, 1996; 48:1180–1195.

[22] A. Branquinho, F. Marcellan, and A. Mendes, Relative asymptotics for orthogo-nal matrix polynomials, Linear Algebra and Its Applications, 2012; 437 n. 7:1458-1481.

[23] N. Ya. Vilenkin et al., Functional Analysis, Wolters-Noordhoff, The Netherlands,1972.

1Department of Mathematics, School of Technology and Management, Poly-technic Institute of Leiria, 2411-901 Leiria, [email protected]

102

2CMUC and Department of Mathematics, University of Coimbra, Apartado3008, EC Santa Cruz, 3001-501 Coimbra, [email protected]

,

3CIDMA and Department of Mathematics, University of Aveiro, 3810-193Aveiro, [email protected]

103



Switched symplectic graphs and their 2-ranks

Aida Abiad1, Willem Haemers2

We apply Godsil-McKay switching to the symplectic graphs over F2 with at least63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increasesafter switching. This shows that the switched graph is a new strongly regular graphwith parameters (22ν − 1, 22ν−1, 22ν−2, 22ν−2) and 2-rank 2ν + 2 when ν ≥ 3. Forthe symplectic graph on 63 vertices we investigate repeated switching by computerand find many new strongly regular graphs with the above parameters for ν = 3with various 2-ranks. Using these results and a recursive construction method for thesymplectic graph from Hadamard matrices, we obtain several graphs with the aboveparameters, but different 2-ranks for every ν ≥ 3.

1Department Quantitative Economics, Operations Research, Maastricht Uni-versity, The [email protected]

2Deptartment Econometrics and Operations Research, Tilburg University,The [email protected]

104



Symmetric generalised polygons

John Bamberg1,

The finite classical generalised polygons have as automorphism groups the finiteLie type groups PSL(3, q), PSp(4, q), PSU(4, q), PSU(5, q), G2(q), 4D3(q), 2F4(q)and they exhibit as much symmetry that is possible for such a geometry: distancetransitivity on points and lines. Thus it is natural to ask whether we can characterisethe classical examples by their symmetry. In this talk, we will present highlights ofthe known results on global symmetry of generalised polygons, and give some recentnew results.

1Centre for the Mathematics of Symmetry and Computation, School of Math-ematics and Statistics, The University of Western [email protected]

105



The finiteness threshold width of lattice polytopes

Monica Blanco1, Christian Haase2, Jan Hofmann2, Francisco Santos1

A lattice d-polytope P ⊂ Rd is the convex hull of finitely many points from Zd.The size of P is the number of integer points it contains, and the width of P isthe minimum, over all non-constant integer linear functionals f , of the length of theinterval f(P ). For each d ≥ 3 and each n ≥ d + 1, there exist infinitely many latticed-polytopes of size n.

In [1] we prove that there exists a constant w∞(d), depending solely on d, suchthat all but finitely many d-polytopes of size n have width at most w∞(d). We callw∞(d) the finiteness threshold width. We show that w∞(d) equals the maximumwidth of a lattice hollow (d− 1)-polytope with infinitely many d-dimensional lifts ofthe same size and width. This allows us to prove that d−2 ≤ w∞(d) ≤ O

(d3/2

)and,

more particularly, that w∞(4) = 2 and w∞(5) ≥ 4. Blanco and Santos had alreadydetermined the value w∞(3) = 1 [2].

References

[1] M. Blanco, C. Haase, J. Hofmann and F. Santos, The finiteness thresholdwidth of lattice polytopes, in preparation.

[2] M. Blanco and F. Santos, Lattice 3-polytopes with few lattice points.Preprint, 23 pages, September 2014, revised November 2015, arXiv:1409.6701.Accepted in SIAM J. Discrete Math.

1Department of Mathematics, Statistics and Computation, University ofCantabria, Av. Los Castros s/n, 39005 Santander, Cantabria, [email protected], [email protected]

2Institut fur Mathematik, Freie Universitat Berlin, Takustraße 9, 14195 Berlin,[email protected], [email protected]

106



Cameron-Liebler sets of generators in polar spacesMaarten De Boeck1, Morgan Rodgers2, Leo Storme1, Andrea Svob3

In [1] Cameron and Liebler studied the orbits of the projective groups PGL(n +1, q). For this purpose they introduced line classes in the projective space PG(3, q)with a specific property, which afterwards were called Cameron-Liebler line classes.Many equivalent characterisations of these Cameron-Liebler classes are known, re-lating them to line spreads, to the row space of the point-line incidence matrix, tothe eigenspaces of the Grassmann scheme, ... Next to proving several equivalentcharacterisations, the main problem is the classification problem.

Cameron-Liebler line classes were also introduced for PG(n, q), n ≥ 4 (see [3] foran overview). Recently Cameron Liebler k-classes in PG(2k + 1, q) were introduced([4]) generalising Cameron-Liebler line classes in PG(3, q) to sets of k-dimensionalsubspaces. Cameron-Liebler classes were also defined and classified for finite sets([2]).

In this talk I will discuss the newly introduced Cameron-Liebler sets of generatorsin polar spaces. I will present several characterisations of these Cameron-Liebler sets,which vary dependent on the polar space. Moreover I will present the classificationof Cameron-Liebler sets of generators in polar spaces with a small parameter.

References

[1] P.J. Cameron and R.A. Liebler, Tactical decompositions and orbits of pro-jective groups, Linear Algebra Appl. 46 (1982), 91–102.

[2] M. De Boeck, L. Storme, and A. Svob, The Cameron-Liebler problem forsets, Discrete Math., 339 (2016), 470–474.

[3] K. Drudge, Extremal sets in projective and polar spaces. Ph.D. Thesis, Universityof Western Ontario, 1998.

[4] M. Rodgers, L. Storme and A. Vansweevelt, Cameron-Liebler k-classes inPG(2k + 1, q), Combinatorica Submitted (2015), 15 pp.

1Department of Mathematics, UGent, Krijgslaan 281 - S22, 9000 Gent, [email protected], [email protected]

2Dipartimento di Tecnica e Gestione dei Sistemi Industriali, University ofPadova, Stradella S. Nicola 3, 36100 Vicenza VI, [email protected]

3Department of Mathematics, University of Rijeka, Radmile Matejcic 2, 51000Rijeka, [email protected]

107



Infinite families of non linear MRD codes

Nicola Durante1, Alessandro Siciliano2,

Let Mm,m′(Fq), with m ≤ m′, be the rank metric space of all the m×m′ matriceswith entries in the finite field Fq. The distance between two matrices is the rank oftheir difference. An (m,m′, q; s)-rank distance code is any subset X of Mm,m′(Fq)such that the minimum distance between two of its distinct elements is s + 1. It islinear if it is a linear subspace of Mm,m′(Fq).

It is known that |X| ≤ qm′(m−s) (Singleton-like bound) [3]. When this bound is

achieved, X is called (m,m′, q; s)-MRD code.There are some infinite families of linear MRD codes for all possible values of the

parameters m, m′, q and s (see e.g. [3]). In finite geometry (m,m, q;m − 1)-MRDcodes are known as spreadsets [1].

To the extent of our knowledge the only non-linear MRD codes, that are notspreadsets, are the (3, 3, q; 1)-MRD codes provided in [2]. In this talk, we will reporton a construction of infinite families of non-linear (m,m, q;m − 2)-MRD codes, forq ≥ 3 and m ≥ 3 (see [4]) that generalize the MRD codes in [2].

References

[1] P. Dembowski, Finite Geometries. Springer 1968.

[2] A. Cossidente, G. Marino, F. Pavese, Non-linear maximum rank distancecodes, Des. Codes Cryptogr., DOI 10.1007/s10623-015-0108-0.

[3] Ph. Delsarte, Bilinear forms over a finite field, with applications to codingtheory, J. Combin. Theory Ser. A 25 (1978), 226–241.

[4] N. Durante, A. Siciliano, Non-linear maximum rank distance codes in thecyclic model for field reduction of finite geometries. Submitted (2016).

1Department of Mathematics and applications “R. Caccioppoli ” - Universityof Naples “Federico II ” - Via Cinthia - IT-80126 Napoli (Italy)[email protected]

2Department of Mathematics and Informatics - University of Basilicata - Viadell’ Ateneo Lucano, 10 - IT-85100 Potenza (Italy)[email protected]

108



Extensions of Minkowski’s theorem on successiveminima

Marıa A. Hernandez Cifre1, Martin Henk2, Matthias Henze3

Let K be a 0-symmetric convex body, i.e., a compact and convex set satisfyingthat K = −K, in the n-dimensional Euclidean space Rn, and let Zn denote the integerlattice. The well-known Minkowski 2nd theorem in the Geometry of Numbers (seee.g. [2]) provides optimal upper and lower bounds for the volume of K in terms of itssuccessive minima:

1

n!

n∏

i=1

2

λi(K,Zn)≤ vol(K) ≤

n∏

i=1

2

λi(K,Zn),

and both bounds are best possible. Here, λi(K,Zn) = minλ > 0 : dim(λK∩Zn) ≥ i

is the i-th successive minimum of K with respect to the integer lattice, 1 ≤ i ≤ n,and vol(K) denotes the volume (Lebesgue measure) of K.

Since this important theorem was discovered, many mathematicians have workedon new proofs and extensions of it. We aim to make a brief historical tour on thisinequality and its generalizations. Then we will show new analogs of the theoremfrom two different points of view (see [1]): either relaxing the symmetry condition, orreplacing the volume functional by the surface area. Thus, if we assume for instancethat the centroid lies at the origin, then the following tight inequality can be proved:

n+ 1

n!

n∏

i=1

1

λi(K,Zn)≤ vol(K).

References

[1] M. Henk, M. Henze, and M. A. Hernandez Cifre, Variations of Minkowski’stheorem on successive minima, Forum Math. 28 (2016), 311–325.

[2] H. Minkowski, Geometrie der Zahlen. Teubner, Leipzig-Berlin, 1896, reprintedby Johnson Reprint Corp., New York, 1968.

1Departamento de Matematicas, Universidad de Murcia, Campus de Es-pinardo, 30100-Murcia, [email protected]

2Institut fur Mathematik, Technische Universitat Berlin, Sekretariat MA 4-1,Strasse des 17. Juni 136, D-10623 Berlin, [email protected]

3Institut fur Informatik, Freie Universitat Berlin, Takustrasse 9, 14195 Berlin,[email protected]

109



Production matrices for geometric graphs

Clemens Huemer1, Carlos Seara1, Rodrigo I. Silveira1, Alexander Pilz2

We present production matrices for non-crossing geometric graphs on point setsin convex position, which allow us to derive formulas for the numbers of such graphs.Several known identities for Catalan numbers, Ballot numbers, and Fibonacci num-bers arise in a natural way, and also new formulas are obtained, such as a formula forthe number of non-crossing geometric graphs with root vertex of given degree. Thecharacteristic polynomials of some of these production matrices are also presented.The proofs make use of generating trees and Riordan arrays.

1Departament de Matematiques, Universitat Politecnica de Catalunya, Spainclemens.huemer,carlos.seara,[email protected]

2Department of Computer Science, ETH Zurich, [email protected]

110



The Manickam-Miklos-Singhi Conjecture in PartialLinear Spaces

Ferdinand Ihringer1

Let P be a set of points, L a set of lines and I ⊆ P × L an incidence relation.Define real-valued weight function f : P → R such that

∑p∈P f(p) = 0. The weight

of a line ` is∑

p∈P`f(p), where P` is the set of points p with p I `.

Consider the case that P is the set N = 1, 2, . . . , n and L is the set of all k-subsets of N . In 1988 Manickam and Singhi conjectured that if n ≥ 4k, then thenumber of lines with nonnegative weight is at least

(n−1k−1). They conjectured the same

for the case, where P is the set of 1-dimensional subspaces of an n-dimensional vectorspace V over a finite field with q, and L is the set of k-dimensional subspaces of V .These conjectures are called MMS conjectures.

We will discuss various results for variants of the MMS conjecture for sets, vectorsspaces and partial linear spaces. Furthermore, we will point out connections betweenthe MMS conjecture, so-called Erdos-Ko-Rado theorems and spreads.

1Mathematisches Institut, Justus Liebig University Giessen, Arndtstrasse [email protected]

111



Domination via (min,+) algebra

Marıa Luz Puertas, Sahar Aleid, Jose Caceres

Perfect codes have played a central role in the development of error-correctingcodes theory. A code in a graph is vertex set such that any two vertices in it are atdistance at least 3. If, in addition, every vertex not in the code has a neighbor init, the code is called perfect. Therefore a perfect code is an independent dominatingset a such every vertex not in the set has a unique neighbor in it. The existenceof this type of codes is not guaranteed in every graph and it has been extensivelystudied. For instance it is well known that the cartesian product Pm2Pn of twopaths has no perfect code unless m = n = 4 or m = 2, n = 2k + 1. In thesecases a less demanding construction could be keeping domination and independencebut admitting at most two neighbors, for vertices not in the dominating set. Aset satisfying these conditions is called independent [1, 2]-set and we have shownthat every grid has one of them. Then the question of calculating ı[1,2](Pm2Pn) thecardinal of such a set with minimum size arises.

The calculation of domination parameters in grids has proved to be a difficulttask. Indeed finding γ(Pm2Pn), the cardinal of a minimum dominating set in thegrid, was an open problem for almost 30 years, since it was first studied in 1984 inrelation with the still open Vizing’s Conjecture. An important milestone on the way

to the solution is the upper bound γ(Pm2Pn) ≤⌊ (m+2)(n+2)

5

⌋−4, for m,n ≥ 8, found

in 1992, like the conjecture about that equality is achieved in case m,n ≥ 16. Thisconjecture was finally confirmed in 2011, ending the study of this problem. Meantimedifferent efforts were made to calculate exact values of γ(Pm2Pn), for fixed small mand every n ≥ m.

Among the different techniques used to address this problem, we would like tofocus on a dynamic programming algorithm developed by applying the (min,+) matrixmultiplication. Values of γ(Pm2Pn) for m ≤ 19 and n ≥ m were obtained with thisalgorithm and we have adapted it to compute ı[1,2](Pm2Pn) for fixed m and n ≥ m.The resulting algorithm can be theoretically applied in every grid, however the longrunning time needed make them useful just on grids of small size, in our case m ≤ 13and n ≥ m. On the other hand ı[1,2](Pm2Pn) can be obtained with a quasi-regularpattern in grids with 14 ≤ m ≤ n, resembling the above-mentioned constructionfor the upper bound of the domination number. In addition this pattern explicitlyprovides an independent [1, 2]-set of minimum size.

Department of Mathematics, University of Almerıa,Carretera Sacramento s/n, 04120 Almerıa (Spain)[email protected], [email protected], [email protected]

112



Rectilinear Convex Hull of a Set of Points in 2DCarlos Seara1

In this talk we review the last progress done in the computation of the rectilinearconvex hull of a set S of n points in the plane in general position, RCH(S) for short.Since the RCH(S) is orientation dependent, we show recent optimal algorithms forcomputing and maintaining RCH(S) for a complete rotation of the coordinate systemin O(n log n) time and O(n) space. We also show optimal algorithms for computingthe rotation angle α such that once we have rotated the coordinate axes by angle αthe area of RCH(S) is maxima; this algorithm runs also in O(n log n) time and O(n)space. Moreover, we will illustrate some statistical applications for computing thefitting of a two-joint orthogonal chain to a set of points (cf. [1, 4]).

We show how to extend the notion of rectilinear convex hull to the oriented hull ofa point set S by considering two non-orthogonal directions, and also for k directionswhere k > 2 (cf. [2, 3]). Finally, we consider the computation of the un-orientedrectilinear convex layers (cf. [5]) and the computation of the orientation of the coor-dinate system such that the number of rectilinear layers is minimum. Some 3D openquestions will be illustrated.

References

[1] C. Alegrıa-Galicia, T. Garduno, A. Rosas-Navarrete, C. Seara and J.Urrutia, Rectilinear convex hull with minimum area, Lecture Notes in ComputerScience 7579 (2012), 226–235.

[2] C. Alegrıa-Galicia, D. Orden, C. Seara and J. Urrutia, Optimizing anoriented convex hull with two directions. In XVI Spanish Meeting on Computa-tional Geometry, Barcelona, 2015.

[3] C. Alegrıa-Galicia, D. Orden, C. Seara and J. Urrutia, On the O-hullof planar point sets. In 30th European Workshop on Computational Geometry(EuroCG 2014), March 3-5, Dead Sea, Israel.

[4] J. M. Dıaz-Banez, M. A. Lopez, M. Mora, C. Seara and I. Ventura,Fitting a two-joint orthogonal chain to a point set, Computational Geometry:Theory and Applications 44(3) (2011), 135–147.

[5] C. Pelaez, A. Ramırez-Vigueras, C. Seara and J. Urrutia, On the recti-linear convex layers of a planar point set. In Mexican Conference on Discrete Math-ematics and Computational Geometry, 60th birthday of Jorge Urrutia, November11-15, 2013, Oaxaca, Mexico, pp. 195-202.

1Departamento de MatematicasUniversidad Politecnica de CatalunyaJordi Girona, 108034 Barcelona (Spain)[email protected]

113



Feynman Integrals, Associated Arrangements andTheir Motives

Ozgur Ceyhan1

The connection between quantum field theory and Grothendieck’s theory of mo-tives aims at understanding the number theoretic aspects of Feynman integrals viatheir period interpretations. The arrangements of the smooth quadric hypersurfaces(a.k.a. the graph hypersurfaces) and the certain hyperplane arrangements (i.e., theconfiguration spaces) have been examined in order to understand the Feynman inte-grals respectively in momentum and position space formulations. I will discuss a newone in this talk; arrangements of singular quadrics. I will describe the motives of sucharrangements associated to the Feynman integrals in φ3 theory.

1Faculte des Sciences, de la Technologie et de la CommunicationCampus Kirchberg, Universite du Luxembourg6, rue Richard Coudenhove-KalergiL-1359 [email protected]

114



Isoparametric hypersurfaces in complex hyperbolicspaces

Jose Carlos Dıaz-Ramos1

A hypersurface in a Riemannian manifold is called isoparametric if all its suffi-ciently close parallel hypersurfaces have constant mean curvature. The aim of this talkis to present the classification of isoparametric hypersurfaces in complex hyperbolicspaces.

1Departamento de Xeometrıa e TopoloxıaFacultade de MatematicasUniversidade de Santiago de Compostela.15782 Santiago de [email protected]

115



Hofer’s geometry of the space of curves.

Michael Khanevsky1

Given a curve in a surface, we consider all curves Hamiltonian isotopic to thegiven one. This space admits a metric induced by Hofer’s metric on the Hamiltoniangroup. We present several examples and discuss tools that can be used to study thegeometry of such spaces.

References

1Departement de Mathematiques, Universite Libre de Bruxelles, AvenueFranklin Roosevelt 50, 1050 Bruxelles, [email protected]

116



Moment maps and closed Fedosov’s star products.

Laurent La Fuente-Gravy1

I will present a moment map µ on the space of symplectic connections on a givenclosed symplectic manifold [1]. Given any symplectic connection ∇, one can build aFedosov’s star product ∗∇, [2]. I will show that the moment map µ evaluated at thesymplectic connection ∇ gives the first non trivial term of the trace density for thestar product ∗∇.

Considering closed Kahler manifolds and working only with Kahler potentials, Iwill explain that the problem of finding the zeroes of µ is an elliptic partial differentialequation. On complex tori and complex projective spaces, I will show that part ofthe zero set of µ has the structure of a smooth finite dimensional manifold [4]. I willalso discuss possible obstructions, analogous to Futaki invariants [3], to the existenceof zeroes of µ on Kahler potentials.

References

[1] M. Cahen and S. Gutt, Moment map for the space of symplectic connections.In Liber Amicorum Delanghe, F. Brackx and H. De Schepper (eds.), pp. 27–36,Academia Press Ghent, 2005.

[2] B. V. Fedosov, A simple geometrical construction of deformation quantization,Journal of Differential Geometry 40 (1994), 384–403.

[3] A. Futaki, Asymptotic Chow semi-stability and integral invariants, Int. J. Math.15 (2004).

[4] L. La Fuente-Gravy, Infinite dimensional moment map geometry and closedFedosov’s star products, Ann. Glob. Anal. and Geom. 49 (1) (2016), 1–22.

1Departement de Mathematiques, Universite de Liege, Place du 20-Aout, 7,4000 Liege, [email protected]

117



Lagrangian mean curvature flow of Hopf toriIldefonso Castro1, Ana M. Lerma1, Vicente Miquel2

The mean curvature flow (MCF) of a smooth immersion F0 : Mn → Rm is afamily of immersions F : M × [0, T )→ Rm parameterized by t, and satisfying

dF

dt(p, t) = H(p, t), F0 = F (·, 0), (?)

where H(p, t) denotes the mean curvature vector of Ft(M) at Ft(p) = F (p, t), and[0, T ) is the maximal time interval such that (?) holds. Unless the flow has an eternalsolution (i.e., it is defined for all t), MCF fails to exist after a finite time, giving riseto a singularity.

These singularities are classified depending on the blow-up rate of the second fun-damental form. The so-called Type I singularities are those such that the blow-upof the second fundamental form is best controlled; the remaining singularities areknown as Type II singularities. Moreover, by the Huisken’s monotonocity formula,these Type I singularities look like self-similar contracting solutions after an appro-priate rescaling.

On the other hand, the Huisken’s classical result: if the initial hypersurface isuniformly convex, then its MCF converges to a round point in finite time, and thefact that Lagrangian self-shrinking spheres do not exist, motivates the open questionposed by Neves about finding a condition on a Lagrangian torus in C2, which impliesthat the Lagrangian mean curvature flow (Mt)0<t<T will become extinct at time T and,after rescaling, Mt converges to the Clifford torus (cf. [2]). In fact, in the Lagrangiancontext, the Clifford torus is the most regular example of a compact self-shrinker forLagrangian MCF in complex Euclidean plane.

In this talk we will answer Neves question by describing the evolution by MCF ofa Hopf torus M0 dividing S3(R0) in two components of equal volume. We will alsogive examples of Lagrangian surfaces developing Type II singularities (cf. [1]).

References

[1] I. Castro, A. M. Lerma, and V. Miquel, Evolution by mean curvature flowof Lagrangian spherical surfaces in complex Euclidean plane, arXiv:1603.03229[math.DG].

[2] A. Neves, Recent progress on singularities of Lagrangian mean curvature flow.Surveys in Geometric Analysis and Relativity, ALM 20 (2011), 413–436.

1Universidad de Jaen, Campus Las Lagunillas, s/n 23071 Jaen, [email protected], [email protected]

2Universidad de Valencia, Burjassot, 46100 Valencia, [email protected]

118



The space of oriented spheres as a bridge betweenH3 and R3

Antonio Martınez1

It is well known that minimal surfaces in Euclidean space R3 and flat fronts in H3

admit a holomorphic representation. They also share in common the fact that thereare many interesting global theorems about their geometry and topology.

Despite these similarities, there is no direct geometric link between these twoclasses of surfaces that are immersed in different ambient spaces. The aim of thistalk is to show a geometric construction that associates to a given flat front in H3 apair of minimal surfaces in R3 that are related by a Ribaucour transformation. Thisconstruction is a particular case of a geometric method to associate surfaces in H3 toa pair of surfaces in R3 that are the envelopes of a smooth congruence of spheres.

1Departamento de Geometrıa y Topologıa, Universidad de Granada, 18071Granada, [email protected]

119



Graded geometry in physics and mechanics.

Vladimir Salnikok1

In this talk I will describe various instances of graded and generalized geometry,appearing naturally in theoretical physics (sigma models, gauging, symmetries offunctionals) and classical mechanics. For the first part, the key idea is that onecan reformulate the property of gauge invariance in the language of equivariant Q-cohomology. This permits to exhibit obstructions to gauging using a nice geometricpicture, and describe the symmetries of some sigma models, including the Dirac sigmamodel, which is universal in the space-time dimension 2. The second part is a work inprogress related to a natural generalization of Hamiltonian systems to so-called port-Hamiltonian, that include dissipative systems and interaction. Generalized geometry,and in particular Dirac structures, turn out to be useful in the context.

References

1Universite du Luxembourg, Bureau 106, Batiment G, Campus Kirchberg 6,rue Richard Coudenhove-Kalergi L-1359 [email protected]

120



Pre-symplectic structures and related deformationproblems

Florian Schatz1

A pre-symplectic structure is a closed 2-form of constant rank, e.g. symplecticforms (full rank) or the trivial 2-form (rank 0). Such structures naturally appear inclassical mechanics, for instance in the process of reduction of a system by Hamiltoniansymmetries. Moreover, they are tightly related to coisotropic submanifolds and giverise to interesting foliations. I will report on an ongoing joint project with MarcoZambon (KU Leuven, Belgium), whose goal is to describe the space of all smalldeformations of a given pre-symplectic structure within the space of all pre-symplecticstructures of some fixed rank. The second goal is to relate this description to otherdeformation problems (such as the deformations of foliations and deformations ofcoisotropic submanifolds).

References

1Universite du [email protected]

121



Kerr-Schild vector fields

Jose M. M. Senovilla1,

The aim of this talk is to introduce a generalization of Killing vector fields in thecase of Lorentzian geometry. They are called Kerr-Schild vector fields because theygenerate the so-called Kerr-Schild transformations. They happen to be associatednot only with the metric g of the manifold (M, g), but also with a given field of nulldirections ` by:

£ξg = 2h`⊗ `, £ξ` = m`

where h and m are smooth functions on M .The main results are contained in [1].

References

[1] B. Coll, S.R. Hildebrandt, J.M.M. Senovilla, Kerr-Schild Symmetries,General Relativity and Gravitation 33 (2001), 649–670.

1Fısica Teorica, University of the Basque Country UPV/EHU, Apartado 644,48080 Bilbao (Spain)[email protected]

122



Constant mean curvature surfaces in Riemannianproduct spaces.

Jose M. Manzano1, Francisco Torralbo2.

The geometry and topology of 3-manifolds with non-negative curvature has beenan active field of research during the last century. One of their most insterestingfeatures is the fact that they generally admit compact (without boundary) embeddedminimal surfaces. Many authors have contributed to the study of compact minimalsurfaces in order to understand the geometry and topology of the 3-manifold. We em-phasize the work of Lawson, who showed the existence of embedded compact minimalorientable surfaces of any topological type in the 3-sphere as well as the first knownexamples of double periodic constant mean curvature surfaces in R3. The techniquehe developed is nowadays known as the conjugate Plateau and has been fruitfully usedin the literature.

We will show how the conjugate Plateau construction can be developed in theRiemannian products S2×R and H2×R addressing its limitations and particularities.Then, we will talk about some advances in the understanding of the surfaces withconstant mean curvature (including minimal ones) in those spaces [1, 2].

Finally, we will also describe a different method to produce minimal surfaces in theaforementioned spaces via the generalized Gauss map and their relation with minimalsurfaces in the 3-sphere and anti-De Sitter space [3, 4].

References

[1] J. M. Manzano, J. Plehnert, and F. Torralbo, Compact embedded mini-mal surfaces in S2 × S1. Communications in Analysis and Geometry. To appear.

[2] J. M. Manzano and F. Torralbo, New examples of constant mean curvaturesurfaces in S2 × R and H2 × R. Michigan Math. J., 63(4) (2014), 701–723.

[3] F. Torralbo, A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in H2×R, J. Math. Anal. Appl. 423(2)(2015), 1660–1670.

[4] F. Torralbo and F. Urbano, Minimal Surfaces in S2 × S2, J. Geom. Anal.,25(2) (2015), 1132–1156.

1Dpt. Mathematics, King’s College London, Strand WC2R 2LS London (UK)[email protected]

2Dpt. Basic Sciences, CUD - San Javier, Coronel Lopez Pena s/n 30720 San-tiago de la Rivera (Murcia)[email protected]

123



Local Nilpotency of the McCrimmon Radical of aJordan System

Jose A. Anquela1

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwichesof the corresponding TKK Lie algebras, we prove local nilpotency of the McCrim-mon radical of a Jordan system (algebra, triple system or pair) over an arbitraryring of scalars. As an application, we get that simple Jordan systems are alwaysnondegenerate.

The results are included in a paper with the same title [1], jointly written byTeresa Cortes, Efim Zelmanov, and the author of this poster, that will appear in theProceedings of the Steklov Institute of Mathematics.

References

[1] J. A. Anquela, T. Cortes, E. Zelmanov, Local Nilpotency of the McCrim-mon Radical of a Jordan System, Proc. Steklov Inst. Math. (to appear).

1Department of Mathematics, University of Oviedo, C/ Calvo Sotelo s/n, E-33007 Oviedo (Spain)[email protected]

124



Some results about diametral dimensions

Francoise Bastin1, Loıc Demeulenaere1, Leonhard Frerick2, JochenWengenroth2

This poster presents the main results of [1] and [2]. Firstly, it gives sufficientconditions to have the equality between two diametral dimensions of a Frechet space(for more precision, see [3, 4, 5, 6, 7, 8]). Secondly, it provides some examples of spacesverifying these conditions. Finally, it gives a family of Schwartz – or even nuclear –(non metrizable) locally convex spaces for which the two diametral dimensions aredifferent.

References

[1] F. Bastin and L. Demeulenaere, On the equality between two diametraldimensions, accepted for publication on April 4, 2016, in Functiones et Approxi-matio, Commentarii Mathematici

[2] L. Demeulenaere, L. Frerick and J. Wengenroth, Diametral dimensionsof Frechet spaces, preprint (April 2016)

[3] H. Jarchow, Locally Convex Spaces, Mathematische Leitfaden, Stuttgart, 1981

[4] A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag, Berlin, 1972

[5] T. Terzioglu, Die diametrale Dimension von lokalkonvexen Raumen, Collect.Math. 20 (1969), 49-99

[6] T. Terzioglu, Diametral Dimension and Kothe Spaces, Turkish J. Math. 32(2008), 213-218

[7] T. Terzioglu, Quasinormability and diametral dimension, Turkish J. Math. 37(2013), 847-851

[8] V. Zahariuta, Linear topologic invariants and their applications to isomorphicclassification of generalized power spaces, Turkish J. Math. 20 (1996), 237-289

1Department of Mathematics, University of Liege, 12 Allee de la Decouverte,4000 Liege, [email protected], [email protected]

2FB IV – Mathematik, University of Trier, 19 Universitatring 54286 Trier,[email protected], [email protected]

125



On differential Galois groups of strongly normalextensions

Quentin Brouette1, Francoise Point1

This work deals with strongly normal extensions, introduced by Kolchin [2]. Theseextentions generalise Picard-Vessiot and Weierstrass extensions. Kolchin proved thatthe differential Galois group of a strongly normal extension L of K is isomorphic to analgebraic group over the field of constants CK of K. Most of the well known results onstrongly normal extensions are proved assuming CK is algebraically closed, notably,the correspondence between the intermediate extensions and the algebraic subgroupsof the differential Galois group of L/K.

We consider strongly normal extensions L/K, without assuming that CK is al-gebraically closed. For instance, the cases CK = R or CK = Qp are studied. Ourpurpose is to extend or adapt the results proved by Kolchin to this context. We statea weaker version of the correspondence and we give an example emphasing the factthat the correspondence given by Kolchin fails in that case. Some of our proofs usethe methods of model theory (see for instance [3]).

The results of this work are in the preprint paper [1].

References

[1] Q. Brouette and F. Point, On Galois groups of strongly normal extensions,http://arxiv.org/abs/1512.05998

[2] E. R. Kolchin, Differential algebra and algebraic groups. Pure and Applied Math-ematics, Vol. 54., Academic Press, New York-London, 1973.

[3] A. Pillay, Differential Galois Theory I, Illinois Journal of Mathematics, volume42 (1998), no. 4, 678 – 699.

1University of Mons, Place du Parc 20, 7000 Mons, [email protected], [email protected]

126



Preliminaries to the study of Radon-typetransforms in a symplectic framework

Michel Cahen1, Thibaut Grouy1, Simone Gutt1

The aim of the project is to define and study Radon-type transforms in a symplec-tic framework. The chosen framework consists of symplectic symmetric spaces whosecanonical connection is of Ricci-type. They can be considered as a symplectic ana-logue of the Riemannian symmetric spaces with constant sectional curvature on whichmany Radon transforms have been widely investigated. A broad range of examplescan be found in S. Helgason’s book [1]. These transforms associate to a compactlysupported continuous function on such a space, another function on a class of sub-spaces by mean of the integration with respect to invariant measures. In our cases,the subspaces for integration are the symplectic totally geodesic submanifolds. Theycan be endowed with an invariant measure. The same holds for the orbits of thosesubmanifolds under the action of a natural group.

First of all, we describe all the spaces involved. On the one hand, we pursuethe study of symplectic symmetric spaces with Ricci-type curvature, initiated by M.Cahen, S. Gutt and J. Rawnsley [2]. On the other hand, we prove that the orbits ofthe selected submanifolds are also symmetric spaces. Then, we define the associatedRadon-type transforms thanks to the existence of invariant measures. The next step isto study these Radon transforms through the exponential mapping and to understandthe obstructions of getting an inversion formula with the same method as used in theRiemannian cases (cf. S. Helgason [1]). For some of our chosen spaces, we know thatthe exponential mapping is a diffeomorphism from an open subset of their tangentspace at a base-point onto a dense open subset of themselves.

References

[1] S. Helgason, Integral geometry and Radon transforms. Springer-Verlag, NewYork, 2011.

[2] M. Cahen, S. Gutt and J. Rawnsley, Symmetric symplectic spaces withRicci-type curvature, Conference Moshe Flato 1999 2 (2000), 81–91.

1Departement de Mathematiques, Universite Libre de Bruxelles, CP 218Boulevard du Triomphe, 1050 Bruxelles, [email protected],[email protected],[email protected].

127



Marginally trapped submanifolds in generalizedRobertson-Walker spacetimes

V. L. Canovas1

The concept of trapped surfaces was originally formulated by Penrose for the caseof 2-dimensional spacelike surfaces in 4-dimensional spacetimes in terms of the signsor the vanishing of the so-called null expansions. More generally, and following thestandard terminology in relativity, a codimension two spacelike submanifold is said tobe marginally trapped if its mean curvature vector field is lightlike. In this work weconsider codimension two marginally trapped submanifolds in the family of generalRobertson-Walker spacetimes.

In particular we derive some rigidity results for this type of submanifolds whichguarantee that, under appropiate hypothesis, the only ones are those contained inslices. We also derive some interesting non-existence results for weakly trapped sub-manifolds. In particular, we give applications to some cases of phisical relevance suchas the Einstein-de Sitter spacetime and certain open regions of de Sitter spacetime,including the so called steady state spacetime. Our results will be an application ofthe (finite) maximum principle for closed manifolds and, more generally, of the weakmaximum principle for stochastically complete manifolds.

This is a joint work with Luis J. Alıas and A. Gervasio Colares.

References

[1] L. J. Alıas, V. L. Canovas and A. G. Colares, Marginally trapped subman-ifolds in generalized Robertson-Walker spacetimes, Preprint (2016).

1Departamento de Matematicas, Universidad de Murcia, E-30100 Espinardo,Murcia, [email protected]

128



A first approach to security analysis on networkswith ancilliary points

M. Carriegos-Vieira1, M.T. Trobajo2,

Network analysis has been an active research field in several scientific areas likesocial sciences, computer sciences, physics or mathematics. From a computer scienceperspective, networks we are interested with in this work, can be identify by a graphG = (V,E), where V is the vertex set - individuals that could be computers orother connected devices -and E is the set of links between them. Therefore, twoindividuals are connected if there exists information exchange between them. Thiskind or graphs deals with symmetric binary and null-diagonal adjacency matriceswhich contain relevant information about the structure of the network. Our objectiveis to hide topological features of the network in order to protect it against targetattacks.

This initial study proposes to add ancilliary individuals and connections to theoriginal network by means of simple stochastic algorithms, analyzing variations oncentrality properties in some particular simulated cases.

References

[1] A. Barabsi, A. Rka and J. Hawoong , Mean-field theory for scale-free randomnetworks, Physica A: Statistical Mechanics and its Applications. 272, 1-2 (1999),pp.173-787.

[2] S.P. Borgatti, Dynamic Social Network Modeling and Analysis:Workshop Summary and Papers, R. Breiger, K. Carley, P. Pattison, Eds.(National Academy of Sciences Press, Washington, DC) (2003)

[3] S.P. Borgatti, A. Mehra, D. Brass and G. Labianca, Network Analysis inthe Social Sciences, Science, 323, (2009) 892-895. DOI: 10.1126/science.1165821

[4] U. Brandes and T. Erlebach Network Analysis, Lecture Notes in computerScience, Springer Berlin Heidelberg New York, (2005).

[5] G. Canright and k. Engo-Monsen, Some relevan aspects of network analysisand graph theory, Handbook of network and system administration, (2007), pp.361-424.

1RIASC, University of Leon, Campus de Vegazana s/n. 24073. Leon (Spain)[email protected]

1Mathematics Department, University of Leon, Campus de Vegazana s/n.24073. Leon (Spain)[email protected]

129



Some applications of the non-abelian tensorproduct of Hom-Leibniz algebras

J. M. Casas 1(1), E. Khmaladze2, N. Pacheco Rego3

Hom-Leibniz algebras were introduced in [3] as triples (L, [−,−], αL) consisting ofa vector space L, a bilinear map [−,−] : L × L → L, and a linear map αL : L → Lsatisfying:

[αL(x), [y, z]] = [[x, y], αL(z)]− [[x, z], αL(y)]

for all x, y, z ∈ L.In this note we introduce a non-abelian Hom-Leibniz tensor product, extending the

non-abelian Leibniz tensor product by Gnedbaye [2], and we analyze its properties.We give applications to universal (α-)central extensions of Hom-Leibniz algebras

and Hochschild homology of Hom-associative algebras.In concrete, we construct the universal (α)-central extension, introduced in [1], of

an (α)-perfect Hom-Leibniz algebra by means of the non-abelian Hom-Leibniz tensorproduct

Further, Hom-type version of Gnedbaye’s result relating Hochschild and Mil-nor type Hochschild homology of associative algebras [2], doesn’t hold for all Hom-associative algebras and requires an additional condition.

References

[1] J. M. Casas, M. A. Insua, N. Pacheco Rego, On universal central extensionsof Hom-Leibniz algebras, J. Algebra Appl. 13 (8) (2014), 1450053 (22 pp.).

[2] A. V. Gnedbaye, A non-abelian tensor product of Leibniz algebras, Ann. Inst.Fourier, Grenoble 49 (4) (1999), 1149–1177.

[3] A. Makhlouf,S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl.2 (2) (2008), 51–64.

1Dpto. Matematica Aplicada I, Universidade de Vigo, Campus UniveritarioA Xunqueira, 36005 Pontevedra, [email protected]

2A. Razmadze Math. Inst. of I. Javakhishvili Tbilisi State University, Tama-rashvili Str. 6, 0177 Tbilisi, [email protected]

3IPCA, Dpto. de Ciencias, Campus do IPCA, Lugar do Aldao, 4750-810 VilaFrescainha, S. Martinho, Barcelos, [email protected]

1First and second authors were supported by Ministerio de Economıa y Competitividad, GrantMTM2013-43687-P (FEDER support is included)

130



Plane curves whose curvaturedepends on distance from a point

Ildefonso Castro1, Ildefonso Castro-Infantes2, Jesus Castro-Infantes3

The fundamental existence and uniqueness theorem in the theory of plane curvesstates that a curve is uniquely determined up to rigid motion by its curvature κ givenas a function of its arc-length. However, in most cases such curves are impossible tofind explicitly in practice, due to the difficulty in solving the quadratures appearingin the integration process. David A. Singer considered a different sort of problem(cf. [3]): Can a plane curve be determined if its curvature is given in terms of itsposition?

Singer started to deal with the posed problem by studying the condition κ(r) =

r :=√x2 + y2, but only the very pleasant special case of the classical Bernoulli

lemniscate, r2 = 3 cos 2θ in polar coordinates, was solved explicitly by him (cf. [3]).Probably, the most interesting solved problem in this setting corresponds to the Eulerelastic curves, whose curvature is proportional to one of the coordinate functions, sayκ(x, y) = c y. Motivated by the above question and by the classical elasticae, the firsttwo named authors recently studied the plane curves whose curvature depends ondistance to a line (cf. [1]). We now study the plane curves whose curvature dependson distance from a point (say the origin, and so κ = κ(r)) requiring the computationof three quadratures (cf. [2]). In this way, we find out several interesting new familiesof spiral curves whose intrinsic equations are expressed in terms of elementary func-tions. We are able to get arc-length parametrizations of them and they are depictedgraphically. We also provide new characterizations of some well known curves, likethe Bernoulli lemniscate, the Norwich spiral and its inverse, the anti-clothoid or thecardioid.

References

[1] I. Castro and I. Castro-Infantes, Plane curves with curvature depending ondistance to a line, Diff. Geom. Appl. 44 (2016), 77–97.

[2] I. Castro, I. Castro-Infantes and J. Castro-Infantes, New plane curveswith curvature depending on distance from the origin, Preprint (2016).

[3] D. A. Singer Curves whose curvature depends on distance from the origin, Amer.Math. Monthly 106 (1999), 835–841.

1Departamento de Matematicas, Universidad de Jaen, 23071 Jaen, [email protected],3Departamento de Geometrıa y Topologıa, Universidad de Granada, 18071

Granada, [email protected],[email protected]

131



Umbilicity properties of spacelike codimension twosubmanifolds

Nastassja Cipriani1, Jose Marıa Martın Senovilla2, Joeri Van der Veken2

Let Φ : (S, g) → (M, g) be an isometric immersion of an n-dimensional Rieman-nian manifold into an (n+ 2)-dimensional semi-Riemannian manifold. S is said to beumbilical with respect to a normal vector ξ if the Weingarten operator Aξ associatedto ξ is proportional to the identity. We say S is pseudo-umbilical (ortho-umbilical) ifit is umbilical with respect to the mean curvature H (by its orthogonal ?⊥H).

We find necessary and sufficient conditions for S to be umbilical. These conditionsare given in terms of algebraic properties of the Weingarten operators, or equivalentlyin terms of the trace-free part of the second fundamental form, that we call total sheartensor. We also find that if S is umbilical then the umbilical direction must be uniqueand we determine it explicitly.

If the ambient manifold M is Lorentzian, then S can be pseudo-umbilical andortho-umbilical at the same time. This is possible when H is null, i.e. g(H,H) = 0.We characterize this situation and present an example for n = 2.

This work generalizes to higher dimension and signature the results presented in(cf. [1]).

References

[1] J. M. M. Senovilla, Umbilical-Type Surfaces in Spacetime. In Recent Trendsin Lorentzian Geometry, Springer Proceedings in Mathematics and Statistics, M.Sanchez et al. (eds.), pp. 87–109, Springer Science+Business Media, New York,2013.

1Department of Mathematics, KU Leuven, Celestijnenlaan 200B – Box 2400,BE-3001 Leuven, [email protected]

2Fısica Teorica, Universidad del Paıs Vasco, Apartado 644, 48080 Bilbao,[email protected]

3Department of Mathematics, KU Leuven, Celestijnenlaan 200B – Box 2400,BE-3001 Leuven, [email protected]

132



The Kurosh Problem for Jordan Nil Systems overArbitrary Rings of Scalars

Teresa Cortes1

We show that quadratic Jordan nil algebras of bounded degree, and, more gener-ally, quadratic Jordan nil algebras satisfying a polynomial identity, are locally nilpo-tent. We also extend these results to Jordan pairs and triple systems.

The results are included in a paper with the same title [1], jointly written by JoseA. Anquela, Efim Zelmanov, and the author of this poster, that has appeared recentlyin Journal of Algebra.

References

[1] J. A. Anquela, T. Cortes, E. Zelmanov, The Kurosh Problem for JordanNil Systems over Arbitrary Rings of Scalars, J. Algebra 444 (2015), 313–327.

1Department of Mathematics, University of Oviedo, C/ Calvo Sotelo s/n, E-33007 Oviedo (Spain)[email protected]

133



On the generalization of analitic local loops

Daniel de la Concepcion1

It is widely known that there is an equivalence of categories between analitic localgroups and Lie algebras. The idea of this poster is to explain the extention of thisresult to a wider class of algebraic structures that generalize groups.

To obtain the Lie bracket from a Lie group the conjugation maps have to bedifferentiated. The next natural step is to consider the algebraic structures thatencode conjugation of groups. These structures are called racks. In recent years,it has been shown that the tangent space of analitic local racks is a Leibniz algebra,realizing that in case the original rack comes from a group conjugation, it is actualyl aLie algebra. The problem arises when trying to find an enveloping algebra for Leibnizalgebras that have some geometric interpretation.

In case of the extension to analitic local loops, the tangent space of such structuresare Sabinin algebras. Eventually, someone could consider both generalizations at thesame time and try to find some structure that generalizes group conjugations, but onloops. The problems of dealing with this generalization are the main concern of thisposter.

1Departamento de Matematicas y Computacion, Universidad de La Rioja,Calle Luis de Ulloa s/n [email protected]

134



Weighted Hardy inequalities, real interpolationmethods and vector measures

R. del Campo1, A. Fernandez2, A. Manzano3, F. Mayoral2, F. Naranjo2

When the real interpolation method (·, ·)ρ,q with a parameter function is applied to

the pairs (L1, L∞) and (L1,∞, L∞) of spaces with respect to a positive scalar measure,the result is a Lorentz space Λqϕ. Namely, when the parameter function ρ belongs tothe class Q(0, 1), introduced by Persson in [3], it holds that (cf. [3, Proposition 6.2])

(L1, L∞)ρ,q = (L1,∞, L∞)ρ,q = Λq tρ(t)

. (1)

In [1] we have established interpolation formulae for different pairs of spaces as-sociated to a vector measure, providing in particular the corresponding version of (1)for the case of vector measures (cf. also [2]).

In this paper we continue the research started in [2] and [1], obtaining resultsthat complements those ones. Now we are interested in analyzing the relationshipbetween some conditions on the pair (ρ, q) and the K-spaces obtained by applying(·, ·)ρ,q,K to the couples (L1, L∞) and (L1,∞, L∞) of function spaces associated to thesemivariation of a vector measure, when ρ is merely a positive measurable functiondefined on (0,∞). Note that for such a kind of functions, the equivalence theorem mayfail, unlike it happens when ρ ∈ Q(0, 1). Our approach is based on the relationshipof the pair (ρ, q) with a weighted Hardy type inequality for non-increasing functions.

References

[1] R. del Campo, A. Fernandez, A. Manzano, F. Mayoral and F. Naranjo,Interpolation with a parameter function and integrable function spaces with re-spect to vector measures, Math. Ineq. Appl. 18 (2015), 707–720.

[2] A. Fernandez, F. Mayoral and F. Naranjo, Real interpolation method onspaces of scalar integrable functions with respect to vector measures, J. Math.Anal. Appl. 376 (2011), 203–211.

[3] L. E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986),199–222.

1Dpto. Matematica Aplicada I, Universidad de Sevilla, ETSIA, Ctra. deUtrera Km. 1, 41013, Sevilla (Spain)[email protected]

2Dpto. Matematica Aplicada II, Universidad de Sevilla, ETSI, Camino de losDescubrimientos, s/n, 41092, Sevilla (Spain)[email protected], [email protected], [email protected]

3Dpto. Matematicas y Computacion, Universidad de Burgos, EscuelaPolitecnica Superior, 09001, Burgos (Spain),[email protected]

135



From elliptic functions to modular forms

Ganim, Marıa de las Mercedes1

This work is part of Elliptic Functions and Elliptic Curves, a thesis to obtain theMaster degree at Univ. Nac. Tucuman, Argentina.

From the standpoint of classical complex analysis, this work describes relation-ships between elliptic functions, elliptic curves and modular forms.

For each lattice Ω of C by using the Weierstass functions ℘ and ℘′ one defines anisomorphism between the complex torus associated to Ω and an elliptic curve in CP 2.

It is shown that the space of modular formsM2k is a direct sum of CG2k and thespace of cusp forms of weight 2k that are multiple of the discriminant function ∆ acusp form of minimum weight 12.

From the Fourier Series of G2k at infinity, a normalization of Eisenstein’s functionsis obtained. The coefficients of that development involve Bernoulli numbers and thefunction σ2k−1(n), which is the sum of the (2k−1)-powers of the positive divisors of n.This illustrates the fact that the Fourier coefficients of modular forms are importantarithmetic functions.

References

[1] L. V. Ahlfors, Complex Analysis: an Introduction to the theory of analyticfunctions of one complex variable. Mc Graw-Hill. Second Edition 1996.

[2] G. A. Jones and D. Singerman, Complex Funtions an algebraic and geometricviewpoint. Cambridge University Press. Fifth Edition. 1997.

[3] J. P. Serre, A Course in Arithmetic. Springer-Verlag New York Inc.1973.

[4] A. W. Knapp, Elliptic curves. Princeton University Press, New Jersey.1992.

1Facultad de Ciencias Exactas y TecnologıaUniversidad Nacional de TucumanAvda. Alem 595 1er Piso Dpto H - San Miguel de Tucuman (4000) [email protected]

136



Some ideas to improve the domain of startingpoints for Newton’s method

J. A. Ezquerro, M. A. Hernandez-Veron

We analyse the semilocal convergence of Newton’s method under center conditionson the first Frechet-derivative of the operator involved [2]. We see that we can extendthe known results so far, since we provide different starting points from the pointwhere the first Frechet-derivative is centered (that is the situation usually consideredby other authors [1, 2, 3]), so that the domain of starting points is enlarged forNewton’s method. We illustrate the theoretical results obtained with some example.

References

[1] I. K. Argyros and S. Hilout, On the quadratic convergence of Newton’smethod under center-Lipschitz but not necessarily Lipschitz hypotheses, Math.Slovaca, 63 (2013) 621–638.

[2] J. M. Gutierrez and M. A. Hernandez, Newton’s method under weak Kan-torovich conditions, IMA J. Numer. Anal., 20 (2000) 521–532.

[3] J. M. Gutierrez, A. A. Magrenan and N. Romero, On the semilocal con-vergence of Newton-Kantorovich method under center-Lipschitz conditions, Appl.Math. Comput., 221 (2013) 79–88.

Department of Mathematics and Computation, University of La Rioja, CalleLuis de Ulloa, s/n. 26004 Logrono, Spain.<jezquer><mahernan>@unirioja.es

137



Some variants of continuous Newton’s method

J. M. Gutierrez, M. A. Hernandez

In this work we are concerned with some numerical properties of the differentialequation

z(0) = z0, z′(t) = − p(z(t))

p′(z(t)), (1)

where z : R → C and p(z) is a given complex function. This problem is calledcontinuous Newton’s method and can be related with the classical Newton’s methodfor solvingg nonlinear equations [1]. It is well-known ([2], [3]) that their solutions flowto the zeros of the p(z) when t → ∞. We show here that roots of multiplicity m ofthe equation p(z) = 0 are asymptotically stable fixed points of the problem (1) withassociate jacobian matrix given by

(−1/m 0

0 −1/m

).

In addition, we explore some properties of the modified continuous Newton’smethod

z(0) = z0, z′(t) = −k p(z(t))

p′(z(t)), k ∈ N. (2)

Actually we show that if k1 < k2 the solutions of (2) for k2 flow to the roots ofp(z) = 0 in a faster way than the solutions of (2) for k1.

References

[1] J. M. Gutierrez, Numerical Properties of Different Root-Finding AlgorithmsObtained for Approximating Continuous Newton’s Method, Algorithms 8 (2015),1210–1218.

[2] J. Jacobsen, O. Lewis and B. Tennis, Approximations of continuous Newton’smethod: An extension of Cayley’s problem, Electron. J. Diff. Equ. 15 (2007), 163–173.

[3] J. W. Neuberger, Continuous Newton’s method for polynomials, Math. Intell.21 (1999), 18–23.

Department of Mathematics and Computer Sciences,University of La Rioja, Logrono, [email protected], [email protected]

138



Dynamical study of the Secant method

J. Manuel Gutierrez1, A. Alberto Magrenan2,

In this work we present a first study on the dynamical behavior of the Secantmethod in order to approximate a solution of a nonlinear equation. Using the tooldeveloped in [1] and the adaptation of the complex dynamics tool it is easy to studythat behavior of that method when it is applied to different functions. The main ideaof the study is to consider both axis as the possible values of the two starting pointsx−1 and x0 and we colour the distinct behaviors, such us convergence to the roots,divergence to infinity, convergence to cycles, etc. in different colors.

References

[1] A. A. Magrenan, A new tool to study real dynamics: The convergence plane Appl.Math. Comput, 248 (2014), 215–224.

1Departamento de Matemticas y ComputacionUniversidad de La RiojaC/ Luis de Ulloa s/n. 26004 Logrono, [email protected]

2Universidad Internacional de La Rioja, Escuela de Ingenierıa.Av. Gran Vıa Rey Juan Carlos I, 41, 26002 Logrono, La Rioja, [email protected]

139



A capture model of Jupiter’s irregular moons basedon a restricted 2 + 2 body problem.

W. Kanaan, V. Lanchares(1),

Giant planets, like Jupiter, have a large number of moons, most of them of smallsize and exhibiting an irregular behavior. Indeed, they move, mainly, in retrogradeorbits with moderate eccentricities. These facts suggest a different origin than theregular moons, greater in size and orbiting the planet in almost circular progradeorbits, close to the equatorial plane. The most feasible explanation is that irregularmoons formed in another place of the Solar System and then captured by giant planets.To explain the capture mechanism, several models have been proposed. Most of themare similar, but with slight differences from one another. In this presentation wepropose a model based on the so called chaos assisted capture. In particular, weconsider a restricted 2 + 2 body problem with two primaries revolving in circularorbit and two small masses do not affect the motion of the primaries. While one ofthe small bodies is temporally captured in a chaotic layer by the giant planet, theother one acts as an intruder. If they are close enough, in their motion around thegiant planet, an energy exchange could take place that permanent captures one of thebodies. Numerical simulations show that the capture is possible and that the mostlikely orbits to be captured are those with retrograde motion.

1Dpto. Matematicas y Computacion Universidade de La Rioja, 26004 Logrno,La Rioja, [email protected], [email protected]

140



From quantum moment maps to deformationquantizations on bounded symmetric domains

Stephane Korvers1

This poster will introduce an explicit construction leading to a realization of the spaceof all invariant deformation quantizations on an arbitrary bounded symmetric domainof Cn. It unifies existing methods giving such deformation quantizations. The methodused in this work is lying at a crossing point between Lie theory, harmonic analysisand non-commutative geometry. It relies on the resolution of hierarchies of PDE’swhich are intimately related with the geometric structure of these domains. Thisapproach extends new methods initiated by Bieliavsky and his collaborators in the2000’s.

References

[1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer,Deformation theory and quantization I & II, Annals of Physics 111 (1978), 61–110 & 111–151.

[2] M. Bertelson, M. Cahen, S. Gutt, Equivalence of star-products, Classicaland Quantum Gravity 14 (1997), A93–A107.

[3] M. Bertelson, P. Bieliavsky, S. Gutt, Parametrizing equivalence classes ofinvariant star-products, Letters in Mathematical Physics 46 (1998), 339–345.

[4] P. Bieliavsky, S. Detournay, Ph. Spindel, The deformation quantizationsof the hyperbolic plane, Communications in mathematical physics 289 (2009),529–559.

[5] P. Bieliavsky, V. Gayral, Deformation quantization for actions of KahlerianLie groups, Memoirs of the American Mathematical Society 236 (2015).

[6] P. Bieliavsky, S. Korvers, The deformation quantizations of bounded sym-metric domains, in preparation (2016).

[7] P. Bieliavsky, M. Massar, Oscillatory integral formulae for left-invariant star-products on a class of Lie groups, Letters in Mathematical Physics 58 (2001),115–128.

[8] S. Gutt, J. Rawnsley, Natural Star Products on Symplectic Manifolds andQuantum Moment Maps, Letters in Mathematical Physics 66 (2003), 123–139.

[9] B. V. Fedosov, A simple geometrical construction of deformation quantization,Journal of Differential Geometry 40 (1994), 213–238.

[10] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Aca-demic Press, New-York, 1978.

141

[11] A. W. Knapp, Lie groups beyond and introduction, 2sd edition. Birkhauser,Boston - Basel - Berlin, 2002.

[12] S. Korvers, Quantification par deformation formelles et non formelles de laboule unite de Cn, Ph.D. thesis, Universite catholique de Louvain (2014).

[13] S. Korvers, Quantum moment maps and retracts for symmetric bounded do-mains, preprint (2015).

[14] I. I. Pyatetskii-Shapiro, Automorphic functions and the geometry of classicaldomains. Gordon and Breach, New-York - London - Paris, 1969.

[15] P. Xu, Fedosov star-products and quantum momentum maps, Communicationsin mathematical physics 197 (1998), 167–197.

1Unite de Recherche en Mathematiques,Universite du Luxembourg,6, rue Richard Coudenhove-KalergiL-1359 Luxembourg, Grand-Duche de [email protected]

142



Superstable expansions of (Z,+, 0)

Quentin Lambotte1,

In [1] and in [2], it is proved that the structure (Z,+, 0,Πq) is superstable, whereΠq is interpreted as the set of powers of a fixed natural number q ≥ 2. In this work,we generalize this result to a wider class of sequences of natural numbers. Specifically,we prove that under one of the following two conditions on a sequence R = (rn)n∈N

of natural numbers, the structure (Z,+, 0, R) is superstable:

1. limn→∞ rn+1/rn = ∞ and R is ultimately periodic modulo m, for any naturalnumber m > 1;

2. R follows a linear reccurence relation and there exists a positive real number θsuch that limn→∞ rn/θ

n ∈ R\0.

Our strategy is the following. We first give a universal axiomatization of the (firstorder) theory of (Z,+, 0, R) in a reasonable language so as to express the behaviourof equations satisfied by elements of R. We then prove that this theory is model-complete and, since our axiomatization is universal, deduce quantifier elimination. Acounting of types argument then shows superstability.

References

[1] B. Poizat, Supergenerix, Jounal of Algebra 404 (2014), 240–270.

[2] D. Palacin and R. Sklinos, On superstable expansions of free abelian groups,preprint (2014),arXiv:1405.0568 .

1Departement de mathematiques, Universite de Mons, Place du Parc 20, 7000Mons, [email protected]

143



Finite metric planes and related objects

Jesse Lansdown1

In the spirit of Hilbert’s axioms (cf. [4]), metric planes further generalise planargeometry through axioms solely relating to incidence, perpendicularity and reflec-tions. The axioms allow for geometries beyond simply the Euclidean plane, includinghyperbolic and elliptic geometries. However, Bachmann (cf. [1]) showed that in thefinite case a metric plane is a Desarguesian affine plane of odd order, and conversely.Sherk (cf. [5]) generalised this result to characterise the finite affine planes of oddorder by removing the ‘three reflections axioms’ from a metric plane. We considerthese and the related partial Sherk planes (cf. [2]) with finitely many points. We willalso show that partial Sherk planes characterise the Bruck nets (cf. [3]) of even degree.

References

[1] F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff. Springer-Verlag, Berlin-New York, 1959.

[2] J. Bamberg, J. B. Fawcett, and J. Lansdown, Bruck nets and partial Sherkplanes, Preprint, arXiv:1601.07231 [math.CO].

[3] R. H. Bruck, Finite nets. II. Uniqueness and imbedding, Pacific J. Math. 13(1963), 421–457.

[4] D. Hilbert, Grundlagen der Geometrie. B. G. Teubner Verlagsgesellschaft mbH,Stuttgart, 1999

[5] F. A. Sherk, Finite incidence structures with orthogonality, Canad. J. Math. 19(1967), 1078–1083.

1Centre for the Mathematics of Symmetry and Computation, School of Math-ematics and Statistics, The University of Western Australia, 35 Stirling High-way, Crawley 6009, Western Australia, [email protected]

144



A boundedness Nikodym property in algebras ofJordan measurable sets

Salvador Lopez Alfonso1,

Let ba(A) be the Banach space of the real (or complex) finitely additive measuresof bounded variation defined on an algebra A of subsets of Ω endowed with thenorm variation. A subset B of A has property N if a B-pointwise bounded subsetM of ba(A) is bounded in ba(A). B has property sN if for each increasing countablecovering (Bm)m of B there exists Bn which has property N and B has property wN ifgiven the increasing countable coverings (Bm1

)m1of B and (Bm1,m2,...,mpmp+1

)mp+1of

Bm1m2...mp , for each p,mi ∈ N, 1 ≤ i ≤ p+ 1, there exists a strand (Bn1n2...nr , r ∈ N)consisting of sets which have property N . The algebra of finite and co-finite subsetsof N fails to have property N and Schachermayer proved that the algebra J (I) ofJordan measurable subsets of I := [0, 1] has property N and J (I) is not a σ-algebra.Valdivia proved in 2013 that the algebra J (K) of Jordan measurable subsets of acompact k-dimensional interval K := Π[ai, bi] : 1 ≤ i ≤ k in Rk has property sN .We have proved that the algebra J (K) of Jordan measurable subsets of a compactsubset K of a metric space has property wN . This result extends Schachermayer andValdivia theorems and enables to give some applications to bounded vector measures.

References

[1] J. Kakol, M. Lopez-Pellicer, On Valdivia strong version of Nikodym boundednessproperty, preprint.

[2] S. Lopez-Alfonso, On Schachermayer and Valdivia results in algebras of Jordanmeasurable sets, RACSAM, DOI 10.1007/s13398-015-0267-x.

[3] S. Lopez-Alfonso, J. Mas, S. Moll, Nikodym boundedness property and webs inσ-algebras, RACSAM, DOI 10.1007/s13398-015-0260-4.

[4] M. Lopez-Pellicer, Webs and Bounded Finitely Additive Measures, J. Math. Anal.Appl. 210 (1997), 257–267.

[5] W. Schachermayer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissertationes Math. 214 (1982) 1-33.

[6] M. Valdivia, On certain barrelled normed spaces, Ann. Inst. Fourier (Grenoble)29 (1979), 39–56.

[7] M. Valdivia, On Nikodym boundedness property, RACSAM107 (2013), 355–372.

1CSA Department, Universitat Politecnica de Valencia, Camino de Vera, s.n.,46022 [email protected]

145



Topological groups and spaces C (X) with orderedbases

M. Lopez-Pellicer1, joint work with J.C. Ferrando and J. Kakol

An index set Σ ⊆ NN is boundedly complete if each bounded subset of Σ has anupper bound at Σ. If Σ is unbounded and directed (and if additionally Σ is boundedlycomplete) a base Uα : α ∈ Σ of neighborhoods of the identity of a topological groupG with Uβ ⊆ Uα, whenever α ≤ β with α, β ∈ Σ, is called in [2] a Σ-base (a longΣ-base). The case Σ = NN has been noticed for topological vector spaces under thename of G-base at [1]. If X is a separable, metrizable and not Polish space, the spaceCc(X) has a Σ-base but does not admit any G-base ([2]). Under an appropriate ZFCmodel the space Cc (ω1) has a long Σ-base which is not a G-base ([2]).

In [2] we proved that (i) if G is a topological group with a long Σ-base then everycompact subset of G is metrizable and (ii) that a Frechet-Urysohn topological groupis metrizable if and only if it has a long Σ-base. This result improve the recent resultin [3] stating that a Frechet-Urysohn topological group with G-base is metrizable.

By (i) if Cc (X) has a long Σ-base then every compact subset of Cc(X) is metrizable(i.e., Cc(X) is strictly angelic). Then X is a C-Suslin space, and we get that Cp(X)is angelic by Orihuela’s theorem at [4], whence Cc (X) is also angelic. Also we showin [2] that a Cp (X) space has a long Σ-base if and only if X is countable.

Problem We do not know whether there exists a topological group with a longΣ-base that admits no G-base.

Problem Let X be a separable metric space admitting a compact ordered coveringof X indexed by an unbounded and boundedly complete proper subset of NN thatswallows the compact subsets of X. Is then X a Polish space?

References

[1] Cascales, B., Kakol, J. and Saxon, S. A., Metrizability vs. Frechet-Urysohn prop-erty, Proc. Amer. Math. Soc. 131 (2003), 3623–3631.

[2] Ferrando, J. C., Kakol, J. and Lopez-Pellicer, M. Spaces C(X) with ordered bases.Submitted.

[3] Grabriyelyan, S., Kakol, J. and Leiderman, A., On topological groups with a smallbase and metrizability, Fundamenta Math. 229 (2015), 129-158.

[4] Orihuela, J., Pointwise compactness in spaces of continuous functions, J. LondonMath. Soc. 36 (1987), 143-152.

1IUMPA and Depto. de Matematica Aplicada,Universitat Politecnica de Valencia, Camino de Vera, s.n., 46022 [email protected]

146



Asymptotic behavior of varying discrete Sobolevorthogonal polynomials

J.F. Manas-Manas1, F. Marcellan2, J.J. Moreno-Balcazar1

Discrete Sobolev (or Sobolev type) orthogonal polynomials have been widely stud-ied in the literature (see, for example, the latest survey (cf. [1])). The contribution ofthis poster to this topic comes from the fact of considering a varying Sobolev innerproduct

(f, g)S =

∫f(x)g(x)dµ+Mnf

(j)(c)g(j)(c), c ∈ R,

where µ is a positive measure supported on the real line and its support can be eitherbounded or unbounded, and Mnn is a sequence of nonnegative numbers satisfyinga very general condition about its asymptotic behavior. Asymptotic properties ofthe orthogonal polynomials with respect to the above inner product have been givenfor particular cases of the measure µ, for example see (cf. [2]) for Jacobi case. Now,strongly inspired by the techniques developed in (cf. [3]), we can obtain the Mehler–Heine type asymptotics which is a very relevant asymptotics for this type of discreteSobolev orthogonal polynomials. As expected these Mehler–Heine formulae dependon the size of the sequence Mnn. In addition, other asymptotic behaviors areprovided.

References

[1] F. Marcellan, and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math.33 (2015), 308–352.

[2] J. F. Manas Manas, F. Marcellan, and J. J. Moreno–Balcazar, Asymp-totic behavior of varying discrete Jacobi–Sobolev orthogonal polynomials, J. Com-put. Appl. Math. 300 (2016), 341–353.

[3] A. Pena, and M. L. Rezola, Connection formulas for general discrete Sobolevpolynomials: Mehler–Heine asymptotics, Appl. Math. Comput. 261 (2015), 216–230.

1Departamento de Matematicas, Universidad de Almerıa, Ctra. Sacramentos/n, 04120 La Canada, Almerıa, [email protected], [email protected]

2Departamento de Matematicas, Universidad Carlos III de Madrid, Avenidade la Universidad 30, 28911 Leganes, [email protected]

147



Area estimates for constant mean curvaturesurfaces in E(κ, τ)-spacesJose M. Manzano1, Barbara Nelli2

Simply-connected Riemannian homogeneous 3-manifolds with isometry group ofdimension 4 or 6, different from the hyperbolic space H3, are classified in a 2-parametric family E(κ, τ), with κ, τ ∈ R. Surfaces with constant mean curvatureH ∈ R (H-surfaces in the sequel) in these ambient spaces have been extensively stud-ied in the literature, being some of the most celebrated results in this field are theexistence of a holomorphic quadratic differential, obtained by Abresch and Rosen-berg, and the solution of the Bernstein problem for H-surfaces whose curvature iscritical (i.e., such that 4H2 + κ = 0) by Fernandez and Mira (cf. [1]). The latterconcerns the classification of surfaces which can be regarded as entire graphs in thedirection of a unit Killing vector field in E(κ, τ). The solution is based on a explicitrelation between entire minimal graphs in Heisenberg space Nil3(τ) = E(0, τ) andholomorphic quadratic differentials in the complex plane or the unit disc, namelytheir Abresch-Rosenberg differential.

Though this is widely considered a satisfactory solution to the Bernstein problem,it does not give any insight into the geometry of such entire graphs. In the presentwork (cf. [2]) we deal with the study of the asymptotic growth of the area of H-graphsintersected with extrinsic or intrinsic metric balls when their radii tend to infinity.We will obtain some sharp upper bounds which can also be applied to understandthe area growth of many other distinguished families of H-surfaces in E(κ, τ)-spaces,not necessarily graphs. Finally we will show that the estimates can be improved ifadditional assumptions on the growth of the height of the surface with respect to agiven minimal section are assumed. Some open problems will be also discussed.

References

[1] I. Fernandez and P. Mira, Holomorphic quadratic differentals and the Bern-stein problem in Heisenberg space, Trans. Amer. Math. Soc. 361 (2011), 5737–5752.

[2] J. M. Manzano and B. Nelli, Height and area estimates for constant meancurvature graphs in E(κ, τ)-spaces. Preprint available in arXiv:1504.05239.

1Department of Mathematics, King’s College London, Strand – WC2R 2LSLondon (United Kingdom)[email protected]

2Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Uni-versita dell’Aquila, Via Vetoio Loc. Coppito – 67100 L’Aquila (Italy)[email protected]

148



Orthogonality for generalized Gegenbauer weightfunctions on the ball with an extra term on the

sphere

Clotilde Martınez, Miguel A. Pinar,

Orthogonal polynomials on the unit ball of Rd with respect to generalized Gegen-bauer weight functions have been studied in [1, 3]. Using spherical polar coordinatesthey can be expressed in terms of h-harmonics and classical Jacobi polynomials withvarying parameters. In this work we study a family of mutually orthogonal polyno-mials on the unit ball with respect to an inner product which includes the generalizedGegenbauer weight function plus an additional mass distributed on the sphere. First,connection formulas relating both families of multivariate orthogonal polynomials areobtained. Then, using the representation formula for these polynomials in terms ofh-harmonics, differential properties will be deduced. In this way some previous resultsby the authors are extended (see [2]).

References

[1] C. F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Encyclo-pedia of Mathematics and its Applications 155, Cambridge Univ. Press, (2014).

[2] C. Martınez and M.A. Pinar, Orthogonal polynomials on the unit ball andfourth-order partial differential equations, SIGMA Symmetry Integrability Geom.Methods Appl. 12 (2016), 020,11 pages.

[3] Y. Xu, Orthogonal expansions for generalized Gegenbauer weight function on theunit ball, Modern Trends in Constructive Function Theory, 153-165, Contemp.Math., 661, Amer. Math. Soc., Providence, RI, 2016.

Departamento de Matematica Aplicada, Universidad de Granada, 18071Granada (Spain)[email protected], [email protected]

149



Some sums of powers of Catalan triangle numbersPedro J. Miana1, Hideyuki Ohtsuka2, Natalia Romero3.

In this contribution we consider combinatorial numbers Cm,k, where

Cm,k :=m− 2k

m

(m

k

), m ≥ 1, k ≥ 0,

which unifies the entries of the Catalan triangles Bn,k and An,k (see [2, 4]) for appro-priate values of parameters m and k, i.e., Bn,k = C2n,n−k and An,k = C2n+1,n+1−k.In fact, some of these numbers are the well-known Catalan numbers Cn that is

C2n,n−1 = C2n+1,n = Cn =1

n + 1

(2n

n

), n ≥ 1.

We present new identities for recurrence relations, linear sums and alternatingsum of Cm,k. After that, we check sums (and alternating sums) of squares and cubesof Cm,k and, consequently, for Bn,k and An,k. In particular, one of these equalitiessolves an open problem posed in [1]. We also present some linear identities involvingharmonic numbers Hn and Catalan triangles numbers Cm,k. Finally, in the lastsection new open problems and identities involving Cn are conjectured. These resultsare collected in the preprint [3].

References

[1] J. M. Gutierrez, M.A. Hernandez, P.J. Miana, and N. Romero, New identities inthe Catalan triangle. J. Math. Anal. Appl., 341 (2008), no. 1, 52–61.

[2] P.J. Miana and N. Romero, Moments of combinatorial and Catalan numbers. J.Number Theory, 130 (2010), no. 8, 1876–1887.

[3] P.J. Miana, H. Ohtsuka and N. Romero, Sums of powers of Catalan trianglenumbers. Arxiv 1602.04347, 2016.

[4] L. W. Shapiro, A Catalan triangle. Discrete Math., 14 (1976), 83–90.

1Departamento de Matematicas, Instituto Universitario de Matematicas yAplicaciones, Universidad de Zaragoza, 50009 Zaragoza, [email protected]

2Bunkyo University High School, 1191-7, Kami, Ageo-city, Saitama Pref., 362-0001, [email protected]

3Departamento de Matematicas y Computacion, Universidad de La Rioja,26004 Logrono, [email protected]

150



On a Moser-Kurchatov type method for nonlinearequations

Mara Jesus Rubio1

A new Kurchatov-type method for the approximation of nonlinear equations inBanach spaces is proposed and analyzed. We consider a modification of Moser’smethod [1] applied to the Kurchatov method [4] to avoid the calculation of inverse.Therefore, the proposed method is an inverse free Kurchatov-type method. Anotheradvantage of this method is that it can be also applied to non differentiable operatorsbecause it is defined from divided difference oprators [5]. The convergence analysis ofthis method is carried out using a technique based on relations of recurrence [2].

The conclusion is that the method improves the applicability of both Newton andkurchatov methods [3] having the same order of convergence.

References

[1] O. H. Hald, On a Newton-Moser type method, Numer. Math. 23 (1975), 411–425.

[2] M. A. Hernandez and M. J. Rubio, A new type of recurrence relations for theSecant method, Int. J. Comput. Math. 72 (1999), 477–490.

[3] M. A. Hernandez and M. J. Rubio, On a Newton-Kurchatov-type IterativeProcess, Numer. Funct. Anal. Optim. 37 (2016), 65–79.

[4] V. A. Kurchatov, On a method of linear interpolation for the solution of fun-cional equations, (Russian) Dolk. Akad. Nauk SSSR 198, 3 (1971), 524–526;translation in Soviet Math. Dolk. 12 (1971), 835–838.

[5] F. A. Potra and V. Ptak, Nondiscrete Induction and Iterative Processes. Pit-man, New York, 1984.

1Department of Mathematics and Computation, University of La Rioja, C/Luis de Ulloa s/n 26004 Logrono. [email protected]

151



Remarks on the set of norm-attaining functionalsand differentiability

V. Montesinos1 A. J. Guirao2 V. Zizler 3

We use the smooth variational principle and a standard renorming technique to give ashort direct proof to the classical Bishop–Phelps–Bollobas theorem on the density ofnorm-attaining functionals for weakly compactly generated Banach spaces. Then weshow that a slight adjustment of a known Preiss–Zajıcek differentiability argumentprovides for a simple, useful characterization of individual norms on separable Ba-nach spaces admitting residual sets of norm-attaining functionals in terms of Frechetdifferentiability of their dual norms.

1Universitat Poltecnica de Valencia, [email protected]

2Universitat Poltecnica de Valencia, [email protected]

3University of Alberta, [email protected]

152



Convergent expansions of special functions in termsof elementary functions

Pedro J.Pagola1, Blanca Bujanda1, Jose Luis Lopez1

In literature we can find a variety of expansions (convergent or not) of differentspecial functions. Usually, these expansions are not valid at the same time for smalland large values of variable. In this work, and using a simple technique, we proposenew uniform convergent expansions of several special functions in terms of elemen-tary functions. These expansions are accompanied by error bounds and numericalexperiments showing the accuracy of the approximations.

1Departamento de Ingenierıa Matematica e Informatica, Universidad Publicade Navarra, Campus Arrosadia s/n, Ed. [email protected],[email protected],[email protected]

153



On Topological Exponential Differential Fields

Nathalie Regnault1,

Question: Does Th(R, <, exp,D), the theory of the real ordered field with expo-nentiation and derivation, have a model-completion?

To answer it, we work in a more general setting, dealing with topological fields ofcaracteristic 0 with an exponential subring on which the exponential is continuous.These structures, which we equipp with an exponential derivation D on which thereisn’t any continuity hypothesis, also encompass the p-adics endowed with a valuationand a partially defined exponential.

References

[1] Wilkie,A. Model completeness results for expansions of the ordered field of realnumbers by restricted Pfaffian functions and the exponential function,J. Amer.Math. Soc. 9 (1996), 1051-1094.

[2] Mariaule,N. On the decidability of the p-adic exponential ring [Thesis]. Manchester,UK: The University of Manchester; 2013.

[3] Guzy,N. & Point, F. Topological Differential Fields, Annals of pure and appliedlogic 161 (2010) 570-598.

[4] Macintyre,A. & Marker,D. & van den Dries,L. Logarithmic-exponential Power Se-ries, J. London Math. Soc. (1997) 56 (3): 417-434.

[5] Kuhlmann,S. & Kuhlmann,F.-V. & Shelah,S. Exponentiation in power seriesfields, Proceedings of the American Math. Society, Volume 125, Number 11,November 1997, 3177-3183.

[6] Kirby,J. Exponential Algebraicity in Exponential Fields, Bull. London Math. Soc.42 (2010) 879-890.

[7] Pierce,D. & Pillay,A. A Note on the Axioms for Differentially Closed Fields ofCharacteristic Zero, Journal of Algebra, Volume 204, Issue 1, 1 June 1998, Pages108-115.

[8] Guzy,N. & Point, F. Topological Differential Fields and Dimension Functions, TheJournal of Symbolic Logic Volume 77, Number 4, Dec. 2012.

1Universite de Mons-Hainaut/Universite Libre de [email protected]

154



On linear refinements of geometric inequalities

J. Yepes Nicolas1,

The Brunn-Minkowski inequality is one of the most powerful theorems in ConvexGeometric Analysis and beyond: it implies, among others, very relevant results such asthe isoperimetric and Urysohn inequalities (see e.g. [3, s. 7.2]). It can be summarizedby stating that the volume (the Lebesgue measure in Rn) is (1/n)-concave, i.e.,

vol((1− λ)K + λL

)1/n ≥ (1− λ)vol(K)1/n + λvol(L)1/n,

for all convex bodies K,L and λ ∈ (0, 1).Moreover, it is well-known that this exponent is necessary and further the best

possible that one may expect. However, a classical result by Bonnesen asserts that ifthe convex bodies have a common volume projection onto a hyperplane, then the vol-ume itself is a concave function, which enhances the statement of Brunn-Minkowski’stheorem.

Here we will show that some other classical inequalities such as the Prekopa-Leindler inequality, the Minkowski first inequality or the isoperimetric inequality sharethis linear demeanor (under assumptions on projections/sections) with the Brunn-Minkowski inequality. Moreover, we will show that the above-mentioned behaviorremains true in the setting of the Gauss Space, i.e., the n-dimensional Euclideanspace Rn endowed with the standard gaussian measure, a fact that will allow us toobtain further Brunn-Minkowski type inequalities for the Gauss measure.

The content of this contribution is based on the works [1, 2, 4].

References

[1] A. Colesanti, E. Saorın Gomez and J. Yepes Nicolas, On a linear re-finement of the Prekopa-Leindler Inequality. To appear in Canad. J. Math. DOI10.4153/CJM-2015-016-6.

[2] E. Saorın Gomez and J. Yepes Nicolas, Linearity of the volume. Looking fora characterization of sausages, J. Math. Anal. Appl. 421 (2) (2015), 1081–1100.

[3] R. Schneider, Convex bodies: the Brunn-Minkowski theory. Second edition.Cambridge University Press, Cambridge, 2014.

[4] J. Yepes Nicolas On characterizations of sausages via inequalities and roots ofSteiner polynomials. To appear in Adv. Geom.

1Instituto de Ciencias Matematicas, CSIC-UAM-UC3M-UCM, C/ Nicolas Ca-brera, 13-15, Campus de Cantoblanco UAM, 28049 Madrid, [email protected]

155

List of participants

1. AIDA ABIAD, Maastricht University

2. ARITZ ADIN, Universidad Publica de Navarra

3. EVA MARIA ALARCON DIAZ, Universidad de Murcia

4. MANUEL ALFARO, Universidad de Zaragoza

5. JOSE ANGEL ANQUELA VICENTE, Universidad de Oviedo

6. ALVARO ANTON SANCHO, Universidad Catolica de Avila

7. DIEGO ARANDA ORNA, Universidad de Zaragoza

8. ALBERTO ARENAS, Universidad de La Rioja

9. SARA ARIAS DE REYNA, Universite du Luxembourg

10. ENRIQUE MANUEL ARTAL BARTOLO, Universidad de Zaragoza-IUMA

11. JOHN BAMBERG, The University of Western Australia

12. ELIAS BARO GONZALEZ, Universidad Complutense de Madrida

13. ROBERTO BARRIO, University of Zaragoza

14. FRANCOISE BASTIN, University of Liege

15. HANS BAUMERS, KU Leuven

16. GLENIER LAZARO BELLO BURGUET, ICMAT/ UAM

17. MANUEL BELLO HERNANDEZ, Universidad de La Rioja

18. PILAR BENITO, Universidad de La Rioja

19. LUIS BERNAL-GONZALEZ, Universidad de Sevilla

20. MONICA BLANCO, University of Cantabria

21. JOSE BONET SOLVES, Universitat Politecnica de Valencia

22. QUENTIN BROUETTE, University of Mons

23. NERO BUDUR, KU Leuven

24. BLANCA BUJANDA, UPNA

156

CONTENTS 157

25. MAURICIO CAICEDO, Vrije Universiteit Brussel

26. ANTONIO CAMPILLO, Universidad de Valladolid

27. MARIA JOSE CANTERO, University of Zaragoza

28. PHILIPPE CARA, Vrije Universiteit Brussel

29. TIMOTEO CARLETTI, University of Namur

30. JOSE CARMONA TAPIA, Universidad de Almerıa

31. MIGUEL CARRIEGOS VIEIRA, Universidad de Leon

32. MARIA JESUS CARRO, Universidad de Barcelona

33. DANIEL CASANOVA ORTEGA, Centro Universitario de la Defensa Zaragoza

34. ENRIQUE CASANOVAS, Universite Libre de Bruxelles

35. JEAN-BAPTISTE CASTERAS, Universidad de Barcelona

36. ILDEFONSO CASTRO, University of Jaen

37. ILDEFONSO CASTRO-INFANTES, University of Granada

38. WOUTER CASTRYCK, Ghent University and KU Leuven

39. FINET CATHERINE, UMons

40. THOMAS CAUWBERGS, Universitat Bielefeld

41. OZGUR CEYHAN, University of Luxembourg

42. TOMASZ CIAS, Adam Mickiewicz University

43. NASTASSJA CIPRIANI, Universidad del Paıs Vasco and KU Leuven

44. EMILIE CLETTE, Universite de Liege

45. RAF CLUCKERS, Laboratoire Painleve

46. TERESA CORTES GRACIA, Universidad de Oviedo

47. RUYMAN CRUZ-BARROSO, La Laguna University

48. PABLO CUBIDES KOVACSICS, Universite de Caen

49. GUILLERMO CURBERA, Universidad de Sevilla

50. JAN DE BEULE, Vrije Universiteit Brussel

51. HENDRIK DE BIE, Ghent University

52. MAARTEN DE BOECK, Ghent University

53. ANA BELEN DE FELIPE, BCAM

54. BERNARDO DE LA CALLE YSERN, Universidad Politecnica de Madrid

158 CONTENTS

55. DANIEL DE LA CONCEPCION SAEZ, Universidad de La Rioja

56. PETER DE MAESSCHALCK, Hasselt University

57. LAURENT DE RUDDER, Universite de Liege

58. ALFREDO DEANO, SMSAS, University of Kent

59. ANDREAS DEBROUWERE, Ghent University

60. ANTONIA DELGADO, Universidad de Granada

61. LOIC DEMEULENAERE, University of Liege

62. JOSE CARLOS DIAZ RAMOS, Universidade de Santiago de Compostela

63. CHRISTIAN DOBLER, University of Luxembourg

64. FREDDY DUMORTIER, Hasselt University

65. NICOLA DURANTE, Universita di Napoli “Federico II”

66. CELINE ESSER, Universite Lille 1

67. JOSE ANTONIO EZQUERRO, University of La Rioja

68. DAVID FARRELLY, Utah State University

69. LIDIA FERNANDEZ, Universidad de Granada

70. JAVIER FERNANDEZ DE BOBADILLA, Ikerbasque-BCAM

71. CARLOS GALINDO, Universitat Jaume I

72. MARIA DE LAS MERCEDES GANIM, Universidad Nacional de Tu-cuman

73. DOMINGO GARCIA, University of Valencia

74. JUAN CARLOS GARCIA ARDILA, Universidad Carlos III de Madrid

75. ELS GOETGHEBEUR, Ghent University

76. MANUEL GONZALEZ, Universidad de Cantabria

77. ALFONSO GORDALIZA, Universidad de Valladolid

78. ERLEND GRONG, University of Luxembourg

79. KARL GROSSE-ERDMANN, Universite de Mons

80. THIBAUT GROUY, Universite Libre de Bruxelles

81. JOSE M. GUTIERREZ, Universidad de La Rioja

82. SIMONE GUTT, Universite Libre de Bruxelles

83. MARIA A. HERNANDEZ CIFRE, Universidad de Murcia

84. LUIS JAVIER HERNANDEZ PARICIO, Universidad de La Rioja

CONTENTS 159

85. CLEMENS HUEMER, Universitat Politecnica de Catalunya

86. NGUYEN HUU KIEN, Universite Lille 1

87. SANTIAGO IBANEZ, Universidad de Oviedo

88. DAVID IGLESIAS LOPEZ, Universidad de Murcia

89. FERDINAND IHRINGER, Justus Liebig University Giessen

90. MANUEL INARREA LAS HERAS, Universidad de La Rioja

91. NGUYEN HUU KIEN, Universite Lille 1

92. ENRIQUE JORDA, Universidad Politecnica de Valencia

93. WAFAA KANAAN, Universidad de La Rioja

94. KAREL KENENS, Hasselt University

95. MICHAEL KHANEVSKY, Universite Libre de Bruxelles

96. ANN KIEFER, Universitat Bielefeld

97. ANNA KIRILIOUK, Universite Catholique de Louvain

98. THOMAS KLEYNTSSENS, University of Liege

99. PHILIPP KORELL, TU Kaiserslautern

100. STEPHANE KORVERS, Universite du Luxembourg

101. LAURENT LA FUENTE-GRAVY, Universite de Liege

102. EDGAR LABARGA, Universidad de La Rioja

103. JESUS LALIENA, Universidad de La Rioja

104. QUENTIN LAMBOTTE, UMons

105. VICTOR LANCHARES, Universidad de La Rioja

106. JESSE LANSDOWN, The University of Western Australia

107. QUY THUONG LE, BCAM-Basque Centre for Applied Mathematics

108. ANA MARIA LERMA FERNANDEZ, Universidad de Jaen

109. CHRISTOPHE LEY, Ghent University

110. SALVADOR LOPEZ ALFONSO, Universitat Politecnica de Valencia

111. VERONICA LOPEZ CANOVAS, Universidad de Murcia

112. MANUEL LOPEZ PELLICER, IUMPA, Universitat Politecnica de Valencia

113. RAFAEL LOPEZ SORIANO, Universidad Complutense

114. IGNACIO LUENGO VELASCO, The University of Western Australia

160 CONTENTS

115. MANUEL MAESTRE, Universidad de Valencia

116. ALBERTO MAGRENAN, Universidad Internacional de La Rioja

117. NYS MANON, Universita degli Studi di Torino

118. ANTONIO MANZANO, Universidad de Burgos

119. JOSE MIGUEL MANZANO, Kings College London

120. MIGUEL MARANON, Universidad de La Rioja

121. NATHANAEL MARIAULE, Universite Paris-Diderot

122. JORGE MARTIN-MORALES, Centro Universitario de la Defensa de Zaragoza

123. AMADOR MARTIN-PIZARRO, Institut Camille Jordan C.N.R.S.

124. ANTONIO MARTINEZ, Universidad de Granada

125. CLOTILDE MARTINEZ, Universidad de Granada

126. CONSUELO MARTINEZ LOPEZ, Universidad de Oviedo

127. FELIX MARTINEZ, Universitat Politecnica Valencia

128. ANA MENDES, Instituto Politecnico de Leiria

129. QUENTIN MENET, Universite d’Artois

130. SERGEI MERKULOV, Universite du Luxembourg

131. MONIA MESTIRI, Umons

132. JUDIT MINGUEZ CENICEROS, Universidad de La Rioja

133. SANTIAGO MOLINA BLANCO, CRM

134. FRANCISCO JOSE MONSERRAT DELPALILLO, Universidad Politecnicade Valencia

135. VICENTE MONTESINOS, Universitat Politecnica de Valencia

136. LEANDRO MORAL, University of Zaragoza

137. JULIO-JOSE MOYANO-FERNANDEZ, Universitat Jaume I

138. JOHANNES NICAISE, Department of Mathematics

139. CARMEN NUNEZ, Universidad de Valladolid

140. MERCE OLLE, Universitat Politecnica de Catalunya

141. DAVID ORDEN MARTIN, Universidad de Alcala

142. PEDRO JESUS PAGOLA, Universidad Publica de Navarra

143. JESUS PALACIAN, Universidad Publica de Navarra

144. DANIEL PALACIN, Universitat Munster

CONTENTS 161

145. IRMA PALLARES, Universitat de Valencia / BCAM

146. ANA ISABEL PASCUAL LERIA, Universidad de La Rioja

147. MARIA PE PEREIRA, ICMAT

148. JOSE ANGEL PELAEZ MARQUEZ, Universidad de Malaga

149. ANTONIO M. PERALTA, Universidad de Granada

150. CARLOS PEREZ MORENO, Universidad del Paıs Vasco y BCAM

151. MARIO PEREZ RIERA , Universidad de Zaragoza

152. ALFRED PERIS, Universitat Politecnica de Valencia

153. ARTHUR PEWSEY, University of Extremadura

154. FRANCOISE POINT, UMons

155. PABLO PORTILLA CUADRADO, ICMAT

156. ELISA POSTINGHEL, KU Leuven

157. ADAM PRZESTACKI, Adam Mickiewicz University

158. MARIA LUZ PUERTAS, University of Almerıa

159. ALEXANDER D. RAHM, Universite du Luxembourg

160. NATHALIE REGNAULT, Universite de Mons-Hainaut/Universite Librede Bruxelles

161. NATALIA MARIA REGO REGO, IPCA

162. MARIA TERESA RIVAS RODRIGUEZ, Universidad de La Rioja

163. FRANCISCO RODENAS, Universidad Politecnica de Valencia

164. JOSE ANGEL RODRIGUEZ MENDEZ, Universidad de Oviedo

165. ANTONIO ROJAS LEON, Universidad de Sevilla

166. JEAN-PHILIPPE ROLIN, Universite de Bourgogne

167. LUZ RONCAL, Universidad de La Rioja

168. MAURIZIA ROSSI, Universit du Luxembourg

169. MARIA JESUS RUBIO, Universidad de La Rioja

170. CRISTINA RUEDA, University of Valladolid

171. JOSE PABLO SALAS ILARRAZA, Universidad de La Rioja

172. ALBERTO SALDANA DE FUENTES, Universite Libre de Bruxelles

173. VLADIMIR SALNIKOV, University of Luxembourg

174. ANA MARIA SANZ GIL, Universidad de Valladolid

162 CONTENTS

175. JESUS MARIA SANZ SERNA, Universidad Carlos III

176. FLORIAN SCHATZ, University of Luxembourg

177. MARTIN SCHLICHENMAIER, University of Luxembourg

178. JEAN SCHMETS, University of Liege

179. MARTIN SCHUMANN, University of Luxembourg

180. CARLOS SEARA, Universidad Politecnica de Catalunya

181. JOSE M M SENOVILLA, UPV/EHU

182. JUAN B. SEOANE SEPULVEDA, Universidad Complutense de Madrid

183. TAMARA SERVI, Universite Paris Diderot

184. IVAN SHESTAKOV, University of Sao Paulo

185. BALDUR SIGURDSSON, BCAM

186. BEATRIZ SINOVA FERNANDEZ, University of Oviedo

187. MARCO STEVENS, KU Leuven

188. DRIES STIVIGNY, KU Leuven

189. ANTONIO SUAREZ, University of Sevilla

190. YVIK SWAN, Universite de Liege

191. LUIS FELIPE TABERA ALONSO, Universidad de Cantabria

192. ANTON THALMAIER, Universite du Luxembourg

193. JAMES THOMPSON, University of Luxembourg

194. FRANCISCO TORRALBO TORRALBO, Centro Universitario de la De-fensa - San Javier

195. STIJN TOTH, Universidad de Cantabria

196. MARIA TERESA TROBAJO DE LAS MATAS, Universidad de Leon

197. CAROLINA VALLEJO RODRGUEZ, Universitat de Valencia

198. JASPER VAN HIRTUM, KU Leuven

199. WIM VEYS, KU Leuven

200. ENRIQUE VIGIL, Universidade do Porto (CMUP)

201. SALVADOR VILLEGAS, Universidad de Granada

202. JASSON VINDAS, Ghent University

203. ANDREAS WEIERMANN, Ghent University

204. JEROEN WYNEN WYNEN, Hasselt University

205. PATRICIA YANGUAS, Universidad Publica de Navarra

206. JESUS YEPES NICOLAS, ICMAT

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