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ABSTRACTS

Aharonov DovTechnion - Israel Institute of Technology, Haifa, Israel

e-mail: [email protected]

Minimal area problems and quadrature domains

I will recall some of the background needed for investigating minimal areaproblems with side conditions. I will then explain how Harold Shapiro andmyself were led, while working on these problems, to study the (generalized)Schwarz function and quadrature domains in the complex plane. Also I willtalk about some recent developments in symmetrization theory that enabledAlexander Solynin, Harold Shapiro and myself to settle an old specific openproblem in this area. The general minimal area problem cannot be attackedby the method of symmetrization and is still wide open.

Aizenberg LevBar-Ilan University, Israel


The classical theorem of Rogosinski on partial sums of power series,its generalizations and applications

The well-known theorem of Rogosinski asserts that if the modulus of the

sum of a power series is less than 1 in the open unit disk:∣∣∣∣ ∞∑n=0

anzn

∣∣∣∣ < 1, |z| < 1,

then all its partial sums are less then 1 in the disk of radius 1/2:∣∣∣∣∣k∑

n=0

anzn

∣∣∣∣∣ < 1, |z| < 12,

and this radius is sharp.We present a generalization of this theorem for holomorphic mappings of

the open unit ball into an arbitrary convex domain. Other multidimensionalanalogs of Rogosinski’s theorem, as well as some applications to dynamicsystems, are considered.

The results obtained are joint work with M. Elin and D. Shoikhet.

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Anderson J. MilneUniversity College London, London WC1, UK


The dyadic parameterization of curves (with F. D. Lesley andV. I. Rotar)

A dyadic parametrization of curves is introduced. This enables connec-tions to be made between the tangential or spiralling properties of curves(usually non-rectifiable) and classical theorems of probability. This is veryuseful for ”constructing” examples.

Azarin VladimirBar-Ilan University, Ramat-Gan, Israel


Some characteristics of asymptotic behavior of subharmonic and entirefunctions and their independence

It is known [1, Ch.3, Th.18] that using any total family of asymptoticcharacteristics, we can check that an entire function of finite order is a func-tion of completely regular growth [2, Ch. 2, 3]. However, even total familiesdo not allow one to check whether a function has an H-multiplicator [3] fora strictly ρ-trigonometrically convex function H.

References

1. A. A. Gol’dberg, B. Ya. Levin and I. V. Ostrovski, ”Entire and mero-morphic functions”, in book Encycl.Math.Sci. v.85 (1997), 4-172.

2. B. Ya. Levin, Distributions of zeros of entire functions, AMS, Provi-dence, RI 1980.

3. V. Azarin and V. Giner, Limit sets and multiplicators of entire func-tions, Adv. Sov. Math. (1992), v.11, 251-275.

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Bergweiler WalterMathematisches Seminar, CAU Kiel, Germany


Normal families and exceptional values of derivatives

A classical result of Hayman says that if f is meromorphic in the planeand f and f (k) − 1 have no zeros for some k ≥ 1, then f is constant. Thecorresponding normality result according to Bloch’s heuristic principle is dueto Gu. We shall discuss some generalizations of these results where insteadof assuming that f and f (k) − 1 have no zeros we only assume that thezeros of these functions are of sufficiently high multiplicity. We also discussthe normality of the family of the logarithmic derivatives f ′/f under thehypotheses that f and f (k) have no zeros.

The results are joint work with J. K. Langley.

Bshouty DaoudTechnion - Israel Institute of Technology, Haifa, Israel


Planar Harmonic Mappings

We review some recent results in the field in two different directions:univalent and multivalent mappings with given dilatations and zeros of poly-nomials.

Chang Der-ChenGeorgetown University, Washington D.C., USA


Sub-Riemannian geometry on a step 2k sub-Riemannian manifold

In this talk, we discuss the behavior of sub-Riemannian geodesics on acertain step 2k sub-Riemannian manifold. We compute the length of thesub-Riemannian geodesics between the origin and any other point. We alsocharacterize the number of sub-Riemannian geodesics between the origin andany other point.

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Chuaqui MartinPontificia Universidad Catolica de Chile, Santiago, Chile


Simple curves in Rn and Ahlfors’ Schwarzian derivative

We derive sharp injectivity criteria for mappings f : (−1, 1) → Rn interms of Ahlfors’ definition of the Schwarzian derivative for such mappings.In addition, we study the issues of continuous extendibility of f to [−1, 1]and of extremal behavior, establishing results similar in spirit to theorems ofGehring and Pommerenke.

Cwikel MichaelTechnion - Israel Institute of Technology, Haifa, Israel


Moduli of continuity and K-divisibility of K-functionals

Most of this talk will be a historical survey. Its main aim is to draw at-tention to the important and by now classical K-divisibility theorem of YuriBrudnyi and Natan Krugljak. Apart from having several significant conse-quences in the theory of interpolation spaces, this theorem has implicationswell beyond interpolation theory, e.g. in connection with moduli of continu-ity. It seems to me that the full potential of these implications has yet tobe realized. Various results about estimates for K-divisibility constants willalso be mentioned, including very recent joint work with Yacin Ameur andsome earlier results obtained with Uri Keich.

Derkach VolodymyrDonetsk National University, Donetsk, UKRAINE


Direct and inverse spectral problem for generalized Jacobi matrices

A new class of generalized Jacobi matrices is introduced. Every general-ized Nevanlinna function m(z), which is a solution of an indefinite momentproblem is proved to be the m-function of a unique generalized Jacobi matrix.The method we use is based on the step-by-step Schur process of solving theindefinite moment problem. As a corollary we obtain the following analog of

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the Stone theorem: every cyclic self-adjoint operator in a Pontryagin spaceis unitary equivalent to some generalized Jacobi matrix. Explicit formulaswhich reconsruct the finite generalized Jacobi matrix H from the spectraldata via the trace formulas are found. The results were obtained in a jointwork with M. Derevjagin.

Dym HarryThe Weizmann Institute of Science, Rehovot, Israel


Riccati equations and bitangential interpolation problems

The aim of this talk is to explain the useful role of Riccati equations in theanalysis of bitangential interpolation problems with singular Pick matrices.The talk will be expository and will assume little beyond some elementarycomplex analysis.

Farkas HershelHebrew University, Jerusalem, Israel


The solutions of some Diophantine equations

Questions about the representation of positive integers as a sum of 2 or4 squares go back at least to Fermat and Legendre. Jacobi however solvedthe quatititative result and gave a formula for the number of solutions ofthe Diophantine equations x2 + y2 = n, x2 + y2 + z2 + w2 = n. In thetalk I shall show how the number of solutions of the former equation givesus a formula for the number of solutions of the latter. The tool will be thetheory of logarithmic derivatives of theta functions. As further examples weshall give formulas for the number of solutions to the Diophantine equations:x2 + y2 + 2z2 + 2w2 = n, x2 + y2 + 3z2 + 3w2 = n.

As in the case of Jacobi, the formula will involve a variant of the classicalsigma function which sums the divisors of a positive integer.

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FitzGerald CarlUniversity of California, San Diego, U. S. A.


The image of the sum of slit mappings

Consider a finitely connected plane domain. There is a normalized, ana-lytic mapping of the domain onto the complement of slits in assigned direc-tions. There is another normalized, analytic mapping onto the complement ofslits orthogonal to those of the first mapping. The sum of these mappings isa mapping of the original domain onto the complement of convex sets. Moreinformation is sought about the image domain. Also, a formula is sought forthe normalized slit mapping with the assigned angles. This last slit mappinghas as its domain, the image of the original pair of slit mappings.

Flatto LeopoldBell Labs, New York, U.S.A.


Billiards in an ellipse

The billiard problem was introduced by Birkhoff in his investigations ofHamiltonian systems. For elliptical tables, the problem is integrable and isrelated to a classical geometric problem of Poncelet. Using notions from Rie-mann surfaces, we obtain the invariant measure associated with this problem.This is in turn used to obtain a complete description of all possible motionsfor billiards in an ellipse.

Gevirtz JulianPontificia Universidad Catolica de Chile


Variational methods for first-order univalence criteria

A first-order univalence criterion for a domain D ⊂ C is a condition ofthe form f ′(D) ⊂ R which implies that f is univalent in D; such a criterionis said to be (weakly) sharp if for all ε > 0 there are nonunivalent g on D forwhich g′(D) is contained in the ε-neighborhood of R. Trivial arguments showthat for a given D vast numbers of such sharp criteria exist; finding them

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is, however, a difficult matter. Indeed, the 27-year-old question of John ([1],[3]) about the largest annulus A = z : 1 < |z| < ρ such that f ′(∆) ⊂ A is aunivalence criterion for the unit disk ∆, remains unanswered. Under minimalassumptions about D and R the discovery of such sharp criteria can bereduced to the analysis of corresponding extremal functions f0, i.e., functionsfor which f ′0(D) ⊂ R and f0(a) = f0(b), for two distinct points a, b ∈ ∂D.We have conjectured that if D is smoothly bounded and simply connectedand R is smoothly bounded, simply or doubly connected, and contained in anannulus 0 < r1 < |z| < r2, then for any such extremal f0, f ′0 = TR(B(T−1

D ))),where TD and TR are canonical mappings of the unit disk onto D and (theuniversal covering surface of) R, respectively, and B is a finite Blaschkeproduct. In [2] we established this for the case of ring domains R bounded bytwo curves strictly star-shaped with respect to 0 and drew from this a numberof qualitative conclusions about the corresponding criteria. In that paper weused a variational procedure based on the representation of f0(z) as G(h0(z)),where G is a canonical mapping of the strip S0 = z : |<z| < 1 onto theuniversal covering surface of R, and |<h0(z)| ≤ 1 in D. In this talk we shalldiscuss various aspects of our previous work on first-order univalence criteriaas well as recent improvements, involving the analysis of the second variationand a more general variational method employing strips with vertical slits inaddition to S0, which allow one to extend the results of [2] to many simplyconnected R for which the methods of that paper were not sufficient.

References

1. J. Gevirtz On extremal functions for John constants, J. London Math.Soc. (2) 39 (1989), 285-298.

2. J. Gevirtz, The theory of sharp first-order univalence criteria, J. Anal-yse Math. 64 (1994), 173-202.

3. F. John, A criterion for univalency brought up to date, Comm. PureAppl. Math. 29 (1976), 293-295.

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Gichev VictorOmsk Branch of Sobolev Intsitute of Mathematics, Russia


Maximal ideal spaces of invariant algebras

Let P be a homogeneous space of a compact Lie group G, A be a closedG-invariant subalgebra of the Banach algebra C(P), and M be its maximalideal space. If A is generated by a finite dimensional invariant subspacethen M can be identified with the polynomially convex hull of an orbit ina finite dimensional linear space. The talking concerns the problem of adetermination of M. There is a satisfactory solution for two opposite cases.1) If P=G with G acting by left and right then M is a semigroup and thedesired description exists. 2) There are necessary and sufficient conditionsfor an orbit to be polynomially convex (joint result with I. Latypov). For thegeneric case there are partial results.

Globevnik JosipUniversity of Ljubljana , Slovenia


Analyticity on circles

Let U be the union of a family S of circles in the complex plane and let f bea continuous function on U which extends holomorphically from each circleC in S to the disc bounded by C. When does it follow that f is holomorphicin the interior of U? We will show how some tools from several complexvariables can be used to deal with this problem.

Gol’dberg AnatoliiBar-Ilan University, Israel


On a connection between the number of poles of a meromorphicfunction and the number of zeros its derivatives

Let f be a meromorphic in finite plane function. We prove an estimate ofthe function N(r, f) from above by using a sum of the functions N(r, 1/fk)for some set of k ∈ N without any restriction of mutiplicity of poles.

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Golberg AnatolyBar-Ilan University, Ramat-Gan, Israel


Homeomorphism with mean integral dilatations

One of the basic properties characterizing quasiconformal mappings is thequasi-invariance of the n-module of the families of separating surfaces. Weconsider more general inequalities. Let G and G∗ be two bounded domainsin Rn, n ≥ 2. For a ring domain D ⊂ Rn, denote by ΣD the family of allcompact piecewise smooth (n−1)-dimensional surfaces Σ which separate theboundary components of D. Let Φ be a finite nonnegative function in G

defined for open subsets E of G such that∑m

k=1 Φ(Ek) 6 Φ(E) for any finitecollection Ekmk=1 of nonintersecting open sets Ek ⊂ E. We denote the classof all such set functions Φ by F . Fix the numbers α, β, γ, δ satisfying

n− 1 ≤ α < β <∞, n− 1 ≤ γ < δ <∞,

and assume that there exists a nonempty family of homeomorphisms f :G→ G∗ such that there are two set functions Φ,Ψ ∈ F not depending on fsuch that for each ring domain D ⊂ G the inequalities

Mβα (Σ∗D) ≤ Φβ−α(D)Mα

β (ΣD),

M δγ (ΣD) ≤ Ψδ−γ(D)Mγ

δ (Σ∗D),

hold. Here Mp(ΣD) is the p-module of ΣD and Σ∗D = f(ΣD). The classof such homeomorphisms is denoted by MS(G) (in fact, it also depends onα, β, γ, δ). Our main result on the differential and geometric properties ofhomeomorphisms with mean quasiconformal dilatations is

Theorem 1. Let n − 1 < α < β ≤ n and n − 1 < γ < δ ≤ n orn ≤ α < β < (n − 1)2/(n − 2) and n ≤ γ < δ < (n − 1)2/(n − 2).Then every mapping f ∈ MS(G) possesses the following properties: (a) fand f−1 are ACL (absolutely continuous on lines); (b) f ∈ W p

1,loc(G) withp = max

(γ, β/(β−n+ 1)

); (c) f−1 ∈W q

1,loc(G∗), q = max

(α, δ/(δ−n+ 1)

).

These bounds for the degrees of integrability are sharp. This result isstrengthened for some special subclasses of mappings. For example, whenf is quasiisometric, quasiconformal, etc., (in the last case, we obtain a newcharacterization of quasiconformality).

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Goryainov VictorVolgograd State University, Volzhsky, Russia

e-mail: goryainov [email protected]

Evolution families of analytic functions

The dynamics of various physical processes are described by means of thecomposition of analytic functions. As a case in point, we mention stochasticbranching process theory, which studies evolution of a population or develop-ment of paste-like matter in discrete or continuous time. On the assumptionof homogeneity, the dynamic of a process with discrete time is describedby iterates of an analytic function. A homogeneous process with continu-ous time is connected with the corresponding one-parameter semigroup ofanalytic functions. The question of embedding analytic functions into con-tinuous semigroups of fractional iterates arises in the study of embeddinghomogeneous processes with discrete time into homogeneous processes withcontinuous time. Recently, the problem of fractional iteration for a gen-eral semigroup P composed of holomorphic mappings f of the unit diskD = z ∈ C : |z| < 1 into itself was solved in various terms in [1]. Wesay that f ∈ P is embeddable if there exists a family f tt≥0 in P such thatf 0(z) = z, f1(z) = f(z), f t+s(z) = f t f s(z) for s, t ≥ 0, and f t(z) → zlocally uniformly with respect to z ∈ D as t → 0. In other words, t → f t

is a one-parameter semigroup in P. The following theorem shows whichrestrictions on f are followed to be embeddable.

Theorem. Let f(z) = c1z+ c2z2 + . . . ∈ P be embeddable and different from

a Mobius transformation. Then the point (c1, c2) belongs to the set:

(w1, w2) ∈ C2 : w1 = e−ξ, w2 = 2w1(1− w1)Re ξ|ξ|

η,

Reξ > 0, |η| ≤ 1.

It may be noted that for an arbitrary f(z) = c1z + c2z2 + . . . in P the

coefficients c1, c2 satisfy the following inequalities

|c1| ≤ 1, |c2| ≤ 1 − |c1|2.

Let S be a semigroup of analytic functions. We say that a subsetwt,s : 0 ≤ s ≤ t ≤ T of S is an evolution family in S if the following con-ditions hold:

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(i) wt,t(z) = z for 0 ≤ t ≤ T ;

(ii) wt,s(z) = wt,r wr,s(z) for 0 ≤ s ≤ r ≤ t ≤ T ;

(iii) wt,s(z)→ z locally uniformly with respect to z ∈ D as t, s→ r.

Evolution families naturally arise in the study of nonhomogeneous processeswith continuous time. In contrast to one-parameter semigroups, the propertyof differentiability for an evolution family is more complicated [2]. We studyconditions which guarantee the differentiability of evolution families. Thepoint is that these families could be described by an evolution equation.

References

1. M.Elin, V. Goryainov, S.Reich, D.Shoikhet, Fractional iteration andfunctional equations for functions analytic in the unit disk (to appear).

2. V. V. Goryainov, Some analytical properties of time inhomogeneousMarkov branching processes, Z. Angew. Math. Mech. 76, Suppl.3(1996), 439-440.

Gromak ValeriiBelarussian State University, Minsk, Belarus


Painleve differential system in complex plane

We discuss the Painleve property and Backlund transformations for thenonlinear ordinary differential systems and review several important generalproperties of the Painleve systems, such as character of singularities, repre-sentation of solutions, special solutions and the number of poles of solutions.

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Hayman Walter KurtImperial College, London, UK

e-mail:

Zeros of solutions to the functional equations∑m

j=1 aj(z)f(cjz) = Q(z)

We discuss zeros of solutions to the functional equations

m∑j=1

aj(z)f(cjz) = Q(z).

Here 0 < |c| < 1 and Q and the aj are polynomials.

Hinkkanen AimoUniversity of Illinois, USA


Growth estimates for certain analytic functions

We discuss growth estimates for analytic functions in the unit disk suchthat the function and some of its derivatives omit certain values. This isjoint work with W.K. Hayman.

Ishankulov TolibSamarkand State University, Samarkand, Uzbekistan


On the possiblity of continuation of functions from a part of theboundary of a domain to the whole domain as solutions of the

Moisil-Theodoresco system of equations

The problem of describing the functions given on a part of the boundaryof a domain which can be holomorphically continued into the domain hasbeen thoroughly studied. The first result was obtained by V. A. Fock andF. M. Kuni [1] in the one-dimensional case. A generalization of the theoremof Fock-Kuni was obtained in a series of papers for holomorphic functionsin several variables [2]. The problem of continuation of functions given ona part of the boundary of a plane domain to this domain as a solution of ageneralized Cauchy-Riemann system, i.e., as a generalized analytic function

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has been considered in [3]. The natural analog of the Cauchy-Riemann sys-tem in three-dimensional space is the Moisil-Theodoresco (MT) system. Inthis talk we consider the problem of describing the functions given on a partof the boundary of a three-dimensional domain which can be continued tothis domain as a solution of the (MT) system. Our investigation is based onthe analog of the Cauchy integral formula for the (MT) system and a jumpformula for limiting values of the Cauchy-type integral [4]. Continuation forthe solution of the (MT) system to the domain by its values on a part of theboundary is based on constructing the Carleman matrix for the (MT) sys-tem. The notion of Carleman function was introduced by M. M. Lavrentev[5]. The Carleman matrix for the (MT) system was constructed in [6]. Byusing the continuation formula we found necessary and sufficient conditionsfor the extendibility of functions given on a part of a boundary to the domainas a solution of the (MT) system. We prove the Fock-Kuni theorem for the(MT) system.

References

1. Fock V.A., Kuni F.M., Dokl. Acad. Nauk SSSR 127:6 (1959), 1195-1198

2. Aizenberg L.A., Kytmanov A.M., Mat. Sb. (1991), 4, p.490-507.

3. Ishankulov T. Sibirsk. Mat. Zh., 41, 1350-1356 (2000).

4. Bicadze A.V., Boundary values problems for second order elliptic equa-tions, Nauka , Moscow, 1966.

5. Lavrentev M.M., Some ill-posed problems of mathematical phisics,Computer Center of the Sibirian Division of the Russion Academy ofsciences, Novosibirsk (1962).

6. Yarmukhamedov Sh., Izv. Acad. Nauk. Uzb. SSR. Ser. Mat. 6, 34-40(1980).

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Kannai YakarWeizmann Institute of Science, Rehovot, Israel


Real and complex aspects of sub-Riemannian geometry forhypoelliptic operators

The study of second order hypoelliptic operators leads naturally to sub-Riemannian geometry. The associated geodesics correspond to critical pointsof a complex analytic action associated by the symbol.

Katsnelson VictorThe Weizmann Institute, Rehovot, Israel


Rational solutions of the Schlesinger system

Rational solutions of the Schlesinger system will be described.

Klebaner FimaSchool of Mathematical Sciences, Monash University, AUSTRALIA


The most visited region of the Lotka-Volterra system

Lotka-Volterra system is a nonlinear system of ordinary differential equa-tions used to model predator-prey coexistence. Rothe (1985) has derivedasymptotics of the times the system spends above and below its equilibriumas its period increases (”Thermodynamics, real and complex periods of theVolterra model”, Z. Agnew Math. Phys. 36, 395-421). We show that a vastmajority of time one of the populations is exponentially small.

This is joint work with R. Khasminskii.

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Kohr Gabriela and Kohr MirelaBabes-Bolyai University, Cluj-Napoca, Romania

e-mail: [email protected] and [email protected]

Loewner chains in several complex variables

In this talk we discuss classical and recent results concerning the theoryof Loewner chains in several complex variables.

Samuel L. KrushkalBar-Ilan University, Ramat Gan, Israel


The Schwarzian derivative and complex Finsler metrics

In the talk, we discuss the extent to which a conformal map f of thedisk onto a Jordan domain D ⊂ C controls the geometric and conformalinvariants of the complementary domain D∗ = C \ D. By controlling, wemean the sharp quantitative distortion of those invariants. The answer isgiven in terms of the Schwarzian derivative Sf = (f ′′/f ′)′ − (f ′′/f ′)2/2 of theRiemann mapping function f and involves its Grunsky coefficients αmn(f).

Let ∆ = |z| < 1, ∆∗ = C \ ∆; B denote the complex Banachspace of hyperbolically bounded holomorphic functions ϕ in ∆∗ with ‖ϕ‖ =sup∆∗(|z|2 − 1)2|ϕ(z)|; A1 = A1(∆) be the Bergman space in ∆, and letA2

1 ⊂ A1 denote the subset of the squares of integrable holomorphic functionsin ∆. One has the pairing 〈µ, ψ〉∆ =

∫∫Dµψdz ∧ dz, where µ ∈ L∞, ψ ∈ L1.

Define for f(z) = z +∞∑1bnz−n ∈ Σ its Grunsky constant

(f) = sup∣∣ ∞∑m,n=1

√mn αmnxmxn

∣∣taking the supremum over x = (xn) ∈ l2 with ‖x‖ = 1. Our key result is:

Theorem 1. Suppose that ‖ϕ‖B < 2 (consequently, the solutions of theSchwarzian equation Sf = ϕ are conformal maps f : ∆∗ → C having quasi-conformal extensions across the unit circle S1). Then the following propertiesare equivalent:

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(i) the Schwarzian derivative ϕ = Sf satisfies

sup‖ψ‖A1(∆)=1

|〈νϕ, ψ〉∆| = sup‖ψ‖

A21(∆)=1

|〈νϕ, ψ〉∆|,

where νϕ(z) = 12(1−|z|2)2ϕ(1/z)1/z4 is the corresponding harmonic Beltrami

coefficient of the Ahlfors-Weill extension of f on ∆;(ii) the extremal (minimal) dilatation k0(f) = infk(wµ) = ‖µ‖∞ :

wµ|∆∗ = f of quasiconformal extensions of f equals its Grunsky constant:k0(f) = (f).

The proof of this theorem relies on the complex metric geometry of theuniversal Teichmuller space T and on some results about the holomorphiccurvatures of complex Finsler metrics. Another main tool is provided by theGrunsky inequalities.

Theorem 1 allows us to solve some problems about the quantitative esti-mation of the Fredholm eigenvalues and of the reflection coefficients of qua-sicircles as well as on extremal quasiconformal extensions, their uniqueness,etc.

The exact bound for an admissible range of the Schwarzians ϕ in Theorem1 arises explicitly in the proof and is determined by the topological structureof the connected component Ω0

ϕ of the intersection of the complex line tϕ :t ∈ C ⊂ B with the universal Teichmuller space T; this component containsthe origin. This approach works well also for more general complex curvesin T.

Kuznetsova OlgaVolgograd State University/ KTH, Stockholm, Sweden


Jacobian of the complex moments mapping: general case

The theme of the talk is the algebraic properties of the complex momentsmapping which arises in the potential analysis of simply-connected domains.Namely, let D be a domain in the complex plane and n is an integer. Thenthe complex moment of D of degree n is the quantity

Mn(D) =1π

∫D

(x+ iy)ndxdy.

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This sequence is well-known in the potential analysis (in particular, it is adiscrete form of the Cauchy transform of D). On the other hand, it concernsthe Hele-Shaw problem of the initial liquid cell with the polynomial datawhen D is a quadrature domain of a kind D = P (U) where P is a polynomialand U is the unit disk. It is well-known that in the last case, the sequenceMk(P ) is finite and gives a mapping of the coefficients body into Cn,n = degP . We discuss the explicit formulae for the Jacobian of this mappingin the case when D = P (U) and extend in various directions the conjecturedUllemar formula for the Jacobian. Our main result provides the expressionfor det J in terms of the roots of the derivative P ′(z). Acknowledgement:This work is supported by the G. Gustafsson foundation.

Kytmanov AlexandrKrasnoyarsk State University, Krasnoyarsk, Russia


Removable singularities of CR functions on singular hypersurfaces

The problem of analytic representation of integrable CR functions onhypersurfaces with singularities is treated. The nature of singularities doesnot matter while the set of singularities has surface measure zero. For sim-ple singularities like cuspidal points, edges, corners, etc., the behaviour ofrepresenting analytic functions near singular points is studied.

Lecko AdamTechical University of Rzeszow, Rzeszow, Poland


On univalent functions starlike with respect to a boundary point

The object of the talk is to present some recent results concerning theclass of univalent functions starlike with respect to a boundary point. Amethod based on Julia’s lemma and on properties of a dynamical systembuilt for the class considered is developed.

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Levin GenadiHebrew University, math. inst., Givat Ram, Jerusalem, Israel


A straightening theorem for a class of meromorphic functions

The Straightening Theorem of Douady and Hubbard states that polynomial-like maps are quasi-conformally conjugate to polynomials. We prove a simi-lar result for maps with a singular point, which are limits of polynomial-likemaps. Corresponding conjugate maps are transcendental meromorphic func-tions. Joint work with Greg Swiatek.

Liflyand ElijahBar-Ilan University, Ramat-Gan, Israel


Hausdorff operators on the real Hardy space

Each integrable function generates a Hausdorff operator. It turns out thatsuch operators are bounded taking the real Hardy space into itself. Variousproperties of these operators are studied, first of all those related to theHilbert transform and to the Fourier transform.

Maergoiz LevKrasnoyarsk State Academy of Architecture and CivilEngineering, Russia


Optimal estimate for extrapolation from a finite set in the Wienerclass

The report is devoted to the problem of analytic extension outside of afinite set from inaccurate data in a Hilbert space H(D) of analytic functionsin a domain D ⊂ Cn.

Let U = z1, ..., zN be a subset of distinct points in a domain D andlet z0 ∈ D \U be a fixed point. We put V = f ∈ H(D) : ||f || ≤ r, wherer > 0. Suppose fj = f(zj) + ζj, j = 1, ..., N are approximate values offunction f ∈ V such that the errors vector ζ = (ζ1, ζ2, ..., ζN) satisfies theassumption: ζ ∈ Sδ = ζ ∈ CN : |ζ| ≤ δ for every f ∈ V. We denote

|ζ|2 =N∑j=1|ζj|2, and δ > 0 is a fixed number.

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For characteristic of quality of extrapolation of any function f ∈ V toz0 ∈ D \ U we consider the optimal error of analytic extension

ΩN(z0; δ) = infEN(z0;α; δ) : α ∈ CN, (1)

where

E2N(z0;α; δ) = sup|f(z0)−

N∑j=1

αjf(zj)|2 +

∣∣∣∣∣N∑j=1

αjζj

∣∣∣∣∣2

: f ∈ V, |ζ| ≤ δ.

(2)

The idea of introduction of this notion is similar to the method of least squaresby K. Miller [1], and it differs from the estimation method by A. A. Melkman,C. A. Micchelli [2]. The main results of the talk were published in [3].

The work was supported by grants of the Russian Foundation for BasicResearch no. 00-15-96140, 03-01-00460.

References

1. Miller K. Least squares methods for ill-posed problems with a pre-scribed bound // SIAM J. Math. Anal, 1970. V. 1, N 1. P. 52-74.

2. Melkman A.A., Micchelli C.A. Optimal estimation of linear operatorsin Hilbert spaces from inaccurate data // Siam. J. Numer. Anal. 1979.V. 16. N 1. C. 87-105.

3. Maergoiz L. S., Fedotov A. M. Optimal discrepancy for analytic exten-sion from a finite set in the case of inaccurate data, for Hilbert spacesof entire functions, Sibirsk. Math. Zh. (2001), Vol. 42, no. 5, pp.1106-1116. (In Russian).

Markina IrinaUniversidad Tecnica Federico Santa Maria, Valparaiso, Chile


Extremal widths on the Heisenberg group

We define the extremal length and extremal width of horizontal vectormeasures associated with sub-elliptic equations on the Heisenberg group. The

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coincidence between the module of the horizontal vector measure system andvarious definitions of capacity is proved. We show the continuity propertyof the module of a special family of curves and reciprocal relations betweenextremal lengths and extremal widths.

Myslivets SimonaKrasnoyarsk State University, Krasnoyarsk, Russia


Holomorphic Lefschetz formula for manifolds with boundary

The classical Lefschetz fixed point formula expresses the number of fixedpoints of a continuous map f : M → M in terms of the transformation in-duced by f on the cohomology of M . The purpose of the talk is to present aholomorphic Lefschetz formula on a compact complex manifold with bound-ary.

Olevskii AlexanderTel Aviv University, Israel


Representation of functions by power series

I will discuss the following problem : which functions on the circle canbe decomposed in a series of exponentials with positive frequencies?

Ournycheva ElenaHebrew University of Jerusalem, Einstein Institute of Mathematics, Jerusalem,

Israele-mail: [email protected]

Radon transforms on matrix spaces

We consider the Radon transform Rf which assigns to a function f(x)on the space Mn,m of real n × m matrices x = (xi,j) the collection of inte-grals over planes in Mn,m. A connection between Rf and Garding-Gindikinfractional integrals associated to the cone of positive definite matrices is es-tablished. By using this connection, we obtain Abel type representations andexplicit inversion formulas for Rf and the corresponding dual transform. It

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is assumed that f belongs to Lp(Mn,m) or the space of continuous functionswith minimal rate of decay at infinity. This is a joint work with Boris Rubin.

Palamodov VictorTel Aviv University, Ramat Aviv, Israel


Darboux equation and reconstruction from spherical means

A construction of the singular fundamental solution for the adjoint Dar-boux equation will be presented. Applications to reconstruction of a functionfrom data of spherical means will be discussed.

Pang XuechengEast China Normal University, Shanghai, P.R. China


Quasinormal families of meromorphic functions omitting aholomorphic function

We study the normality and quasinormality of families of meromorphicfunctions on plane domains. Our main results are the following.

1. Let F be a family of functions meromorphic on the plane domain D,all of whose poles are multiple and whose zeros all have multiplicity atleast 3. Let h(z) (not identically equal to 0 or infinity) be a functionmeromorphic on D. If, for each f ∈ F, f ′(z) 6= h(z) for all z ∈ D, thenF is a normal family on D.

2. Let F be a family of functions meromorphic on the plane domain D,all of whose zeros are multiple. Suppose that for each f ∈ F, f ′(z) 6= 1for all z ∈ D. Then if F is quasinormal on D, it is quasinormal of order1 there.

This is joint work with Shahar Nevo and Larry Zalcman.

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Rabinovich VladimirInstituto Politecnico Nacional, Mexico, D.F., MEXICO


Essential spectrum of Schrodinger operators

We consider the electromagnetic Schrodinger operator

H =n∑j=1

(i∇j − bj(x)I)2u(x) + a(x)I,

where a(x), bj(x) ∈ L∞,rich(Rn), which is considered as unbounded operatorin L2(Rn). Here L∞,rich(Rn) is a subalgebra of functions c ∈ L∞(Rn) withthe following property: every sequence hm →∞ has a subsequence hmk suchthat there exists a strong limit

s− limk→∞

c(x+ hmk)I = ch(x)I,

an operator multiplication by the function c(x) ∈ L∞(Rn). The operator Hh

which obtained from H by changing of a(x), bj(x) to ah(x), bhj (x) is called thelimit operator of H defined by the sequence hmk . We denote by Lim(H) theset of all limit operators of H. The following theorem give a full descriptionof the essential spectrum of H.

Theorem. ess spH =⋃Hh∈Lim(H) spHh.

There are numerous cases in which the limit operators have a structuresimple enough to allow us to give an effective description of essential spec-trum of electromagnetic Scrodinger operators for new interesting classes ofpotentials.

Schiff JeremyBar-Ilan University, Ramat Gan, Israel


A modification of the modified equations method

One of the standard approaches used to compare solutions of differentialequations and the results of numerical simulations is the so-called “modifiedequations method”. This consists of hunting down a different differential

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equation for which the numerical simulation is exact, and then comparingbetween the original differential equation and the modified one. We show thatexcept of the simplest cases, there are qualitative features of the dynamics ofnumerical simulation that the standard modified equations method can sim-ply not capture. We propose a modification in which the modified equationsare not differential but integro-differential, and give a much more accuratepicture of the true behavior of the numerics.

Schreiber BertramWayne State University, Detroit, MI, USA


Representation of isotropic harmonizable covariances

A random field on Rn with finite variances is called isotropic if its meansare rotationally invariant and its covariances are invariant under the naturaldiagonal action of SO(n) on Rn × Rn. The theory of weakly harmonizablerandom fields is well established, while several authors, such as Yaglom,Yadrenko, and Swift, have studied stationary and nonstationary isotropicprocesses and fields. In this work, conducted jointly with M.M. Rao, westudy the spectral theory of isotropic, weakly harmonizable random fieldsand discuss an integral representation for their covariances.

Bert–Wolfgang SchulzeUniversity of Potsdam, Germany


Ellipticity on spaces with corners

The program to analyse elliptic differential (and pseudo-differential) oper-ators on manifolds with conical points, edges, or higher corners, is connectedwith the control of additional symbolic levels. A special case are bound-ary value problems in the interpretation of manifolds with edges where theboundary is the edge and the inner normal the model cone of the correspond-ing local wedges. In the case of ‘higher’ (say, regular) singularities we have ahierarchy of principal symbols, where most of the components are operator-valued. The Fredholm property is suitable scales of weighted Sobolev spacesrequires additional conditions on the strata of lower dimension. We illustrate

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this in the case of manifolds with corners where the base is a manifold withedges. An alternative description is the case of manifolds with edges whohave conical singularities.

References

1. B.–W. Schulze, Operators with symbol hierarchies and iterated asymp-totics. Publ. RIMS, Kyoto University 38, 4 (2002), 735-802.

2. L. Maniccia and B.–W. Schulze, An algebra of meromorphic cornersymbols. Bull. des Sciences Math. 127, 1 (2003), 55-99.

Siciak JozefJagiellonian University, CRACOW, Poland


Pluripolar hulls and pseudocontinuation

We discuss a generalized analytic continuation expressed in terms of theplurisubharmonic hull of the graph of a meromorphic function. In particular,we give an example of a function f holomorphic in the unit disc such that fdoes not have a pseudocontination across any subset E of the unit circle ofpositive measure, while there exists a meromorphic function F in |z| > 1such that:

If U is a function plurisubharmonic on C2 such that U(z, f(z)) = −∞ onthe unit disc then U(z, F (z)) = −∞ for all z with |z| > 1, F (z) 6=∞.

Silverman HerbertCollege of Charleston, South Carolina, USA


Inverses under convolution

We investigate classes of univalent functions for which there exist univa-lent inverses under convolution.

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Sodin MikhailTel Aviv University, School of Mathematical Sciences, Israel


Harmonic functions and sign changes

This is a report on a joint with Leonid Polterovich work in progress.Let u be a harmonic function in the unit disc D continuous up to the

boundary, u(0) = 0, and let N be a number of sign changes of u on the unitcircle. Then

Areaz ∈ D : u(z) > 0 ≥ C

N,

where C is a positive numerical constant. The estimate is sharp up to thechoice of C.

This gives a quantitative version of a result of Nadirashvili. We shall alsodiscuss applications to metric geometry of nodal domains of eigenfunctionsof Laplacian on two-dimensional surfaces.

Tarkhanov NikolaiUniversity of Potsdam, Germany


Holomorphic Lefschetz formula

We show a Lefschetz fixed point formula for holomorphic functions in abounded domain D with smooth boundary in the complex plane. To intro-duce the Lefschetz number for a holomorphic map of D, we make use of theBergman kernel of this domain. The Lefschetz number is proved to be thesum of usual contributions of the fixed points of the map in D and contribu-tions of all boundary fixed points, these latter being different for attractingand repulsing fixed points.

Tkachev VladimirVolgograd State University/ KTH, Stockholm, Sweden


The level-sets analysis of holomorphic functions

We discuss our recent progress concerning the length functions on thegeneralized lemniscates. Let f(z) be a holomorphic function with non-empty

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zero set. Then a compact component Et(f) of the level set |f(z)| = et iscalled the generalized lemniscate of f . The famous Erdos Conjecture askswhether the length of the polynomial lemniscate |P (z)| = 1 is greatest in thecase when P (z) = zn − 1. We obtain the followingTheorem. Given a holomorphic function f(z), the length function |Et(f)|can be represented as the bilateral Laplace transform of a positive measure µf

|Et(f)| =∫ +∞

−∞extdµf (x)

in every regular interval t ∈ ∆, which does not contain singular values of f .In particular, Et(P ) is an exponentially convex function.

We also obtain explicit formulae for the derivatives of the length function|Et(f)| in terms of the associated Hilbert space (which leads to the positivityof the sequence of successive derivatives and to the representation above).Corollary. If the polynomial P is extremal for the Erdos conjecture, it hasa singular point on the lemniscate |P | = 1.

This results refines the recent theorem due to Eremenko & Hayman inMich. Math. J., 1999. We also give a detailed discussion of the structureof the family of close to extremal polynomials (the so-called D-polynomials)which have their singular values on a single lemniscate. In particular, weobtain lower estimates on the length function if the function f has a singularvalue with high multiplicity. Another merit of our approach is that it impliesthe explicit formula (in terms of Gauss’ hypergeometric function) for thelength function when f satisfies the equation

f ′ = (1− fk)m.

We also discuss some aspects of potential and analytic properties of the lengthfunctions in the case of D-polynomials.

Vasil’ev AlexanderUniversidad Tecnica Federico, Valparaiso, Chile


Flows on homogeneous spaces and a parametric method for conformalmaps with quasiconformal extension

We study flows on the universal Teichmueller space T and on the homo-geneous Kirillov space Diff S1/S1 embedded in T . As a result, we construct

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a parametric representation for conformal maps which admit quasiconformalextension and, in particular, such that the associated quasidisks are boundedby smooth Jordan curves. The Douady-Earle extension is used. Some appli-cations to Hele-Shaw flows of viscous fluids are given.

Volkovich Zeev(Vladimir)Ort Braude College, Karmiel, Israel


Semigroups of holomorphic mappings on the unit disk with aboundary fixed point

In this talk we study the class of all holomorphic semigroup generatorson the open unit disk having boundary null points. This class arises, inparticular, in many models of stochastic branching Markov processes.

Further, we give an asymptotic representation of starlike functions ofthe so-called Robertson class. We obtain a criterion of conformality of suchfunctions at boundary null points. This is joint work with M. Elin and D.Shoikhet.

Vodop’yanov SergueiSobolev Institute of Mathematics, Novosibirsk, RUSSIA


Geometry of Carnot-Caratheodory spaces and differentiability ofmappings

We prove cc-differentiability of the Lipschitz mappings on Carnot-Caratheodoryspaces. As a consequence, we obtain some results of geometric measure the-ory and related topics.

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Wang YuefeiAcademy of Mathematics and System Sciences, Chinese Academy of Sciences,

Beijing, Chinae-mail: [email protected]

On the Fatou set of holomorphic maps

We discuss a few results on topological properities of the Fatou compo-nents of certain transcendental holomorphic maps.

Yakubov EduardHolon Academic Institute of Technology, Holon, Israel


The Beltrami equation and finite mean oscillation

Existence and representation theorems for ACL solutions of the Beltramiequation with the dilatation majorized by functions of finite mean oscillationwill be considered. This is a joint work with Vladimir Ryazanov and UriSrebro.

Yomdin YosefThe Weizmann Institute, Rehovot, Israele-mail: [email protected]

Center-Focus Problem, Moments, and Compositions — Some NewDevelopments

The talk will present an overview of the mutual relations between thesubjects mentioned in the title, stressing some new developments:— Center equations as a non-linear deformation of the Moment equations;— Counterexamples to the Composition Conjecture for the one-sided Mo-ments, as well as an ”almost complete” positive answer to this problem,found recently by F. Pakovich;— Geometry of the Center equations versus the Moment equations.

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Yudytskiy PetroMSU, East Lansing, US


Szego, Killip–Simon and others

A major breakthrough has been made in the spectral theory of Jacobimatrices in the case when the spectral measure σ is supported on an intervalE plus a denumerable set of mass points which accumulate at the bound-ary points of E only. Together with Franz Peherstorfer we extended thefamous Szego Theorem on asymptotic for the orthonormal polynomials onthe case when in addition to the Szego’s condition on E the mass points sat-isfy Blaschke’s condition. Sergey Denisov proved the Nevai conjecture, whichextend Rakhmanov’s Theorem. Killip and Simon found an exact conditionon the coefficients of a Jacobi matrix such that σ′ is in Quasi Szego on Eand the mass points in Quasi Blaschke (this result is to appear in Annals ofMathematics). We present a general point of view on results of this kind.

Yger AlainBordeaux 1 University, France


Trace and residue calculus

I will present recent developments of multidimensional residue and tracecalculus, together with applications to inverse problems related to the ge-ometric Radon transform, to division problems in polynomial rings and toarithmetic intersection or division theory.

Zaharyuta VyacheslavSabanci University, Tuzla/Istanbul, Turkey


Lh-potential theory and its applications in harmonic analysis

Lh-potential theory, dealing with so-called Lh-functions and maximalLh-functions, is a proper tool for studying of Hadamard-type inequalities forharmonic functions in Rn. It serves harmonic functions much as complexpotential theory does for analytic functions of several complex variables,dealing with plurisubharmonic and maximal plurisubharmonic functions.

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We discuss some applications to studying of orthogonal harmonic poly-nomials on compact subsets in and related approximation problems for har-monic functions. In that connection, we consider also some natural Lh-analogues of Green function, capacity, transfinite diameter, Chebyshev con-stants for a compact set in Rn .

Zalcman LawrenceBar-Ilan University, Ramat-Gan, Israel


A glance backward, a long look ahead

A selective (and, necessarily, much abbreviated) account of my mathe-matical doings over the past forty years, with an emphasis on some of myfavorite unsolved problems.

Zemanek JaroslavInstitute of Mathematics, Polish Academy of Sciences, Warsaw, Poland


Cesaro means and resolvents of linear operators

This joint work with Juan Sanchez relates the growth of powers and theirCesaro means to the behaviour of the partial sums of the resolvent of aBanach space operator.

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