A VIEW FROM THE CHAIRNicholas Ercolani

Department Head of Mathematics and Professor

ur Mathematics Department has traditionally embraceda broad view of its mission. Faculty members, as well as

students, participate in a wide range of activities that extendoutside of the Department per se. These include contributions tothe profession of Mathematics at all levels, interdisciplinary re-search and service to the community. The present newsletter givesus the opportunity to describe recent developments in some ofthese endeavors.

Many of our faculty participate in cross-disciplinary researchand educational projects through the Department’s unique rela-tion with the Interdisciplinary Program in Applied Mathemat-ics. This issue of the newsletter illustrates two such connections.One is the article by Jim Cushing that describes the interplay between modern DynamicalSystems theory and the modeling of complex biological and ecological systems. Another isthe article by former Applied Mathematics graduate student Annalisa Calini whose researchapplies techniques of nonlinear science to the study of classical problems in DifferentialGeometry. Annalisa is one example of a number of former graduate students in both theMathematics and Applied Mathematics graduate programs whose advisors were Depart-mental faculty and who went on to successful research careers at the interface between pureand applied Mathematics.

Funding from VIGRE and GIG grants as well as other sources support a number of ongoingactivities that enrich the research experiences of our students and faculty. Examples of thisare described in the articles of Jeff Selden, David T. Gay and Doug Ulmer.

Departmental faculty continue to be successful in garnering gifts and awards which enableus to maintain a multifarious array of outreach activities such as enrichment and support ofregional secondary Mathematics teachers, as described in the articles by Fred Stevenson andElias Toubassi, and Program ACCESS whose story is recounted in David Lovelock’s article.

In future issues we will be telling you more about our programs and activities. To ouralumni, I would also like to extend an invitation to write to us. We would be delighted tohear from you and perhaps some of you would even like to share your experiences in thesepages. So please contact us and let us know what you are doing!

Contact us at: http://www.math.arizona.edu/~mcenter/alum/alum.html

In this Issue

PAGE TWOComplex PopulationDynamicsJim Cushing

PAGE SIXCenter for Recruitment &Retention of SecondaryMath TeachersFred Stevenson

PAGE EIGHTEnriching High SchoolMathematicsElias Toubassi

PAGE EIGHTA Myopic View ofProgram ACCESSDavid Lovelock

PAGE NINEOne Year in ChinaDavid T. Gay

PAGE ELEVENA Graduate Experiencein GermanyJeff Selden

PAGE TWELVEThe Southwestern Centerfor ArithmeticalAlgebraic GeometryDouglas Ulmer

PAGE THIRTEENSolitary Smoke RingsAnnalisa Calini

PAGE SIXTEENPresentations, Symposia,Workshops

Mathematics at UADepartment of MathematicsThe Universtiy of ArizonaPO Box 210089Tucson AZ 85721-0089

Nicholas Ercolani

Presentations, Symposia,Workshops

Nicholas Ercolani, Department Head and Professor, is theco-organizer of a Symposium on “Waves: Patterns andTurbulence” for the Annual Meeting of the AmericanAssociation for the Advancement of Science in Boston,February 14-19, 2002.

Dr. Ercolani will be a Plenary Speaker for the AMS SpringEastern Section Meeting in Montreal May 3-5, 2002.

Professor Hermann Flaschka will speak at the workshop onGeometry, Mechanics, and Dynamics, for the Field Institutein Toronto, August 7-11, 2002.

Yi Hu, Associate Professor, is an invited speaker at the ICM2002 Satellite Conference in Algebraic Geometry in Shang-hai, China, August 13-17, 2002.

Professor Moshe Shaked has been invited to give a presenta-tion for the Third International Conference on Mathemati-cal Methods in Reliability, Methodology, and Practice, June17-20, 2002 at the Norwegian University of Science andTechnology in Trondheim.

Doug Ulmer, Associate Professor, is co-organizing the AMS& Italian Mathematical Union, Special Session on Arith-metical Algebraic Geometry, Pisa, Italy June 12-16, 2002.

Professor Maciej Wojtjiwski will be an invited Speaker forthe International Congress of Mathematics 2002 in Beijing,China, August 20-28, 2002.

NON-PROFIT ORG.U.S. POSTAGE

PAIDTUCSON, ARIZONA

PERMIT NO. 190

based on “noisy equilibrium” states. As he put it in hisseminal 1976 paper, the fact that a simple, deterministicequation

“can possess dynamical trajectories which look likesome sort of random noise has disturbing practicalimplications. It means, for example, that apparentlyerratic fluctuations in the census data for an ani-mal population need not necessarily betoken eitherthe vagaries of an unpredictable environment orsampling errors: they may simply derive from a rig-idly deterministic population growth relation-ship.”3

May’s tenet raised the possibility of new ways to under-stand ecological systems. Although unexplained noisewill always be present in ecological data, these insightsprovided a broad new hypothesis concerning the role ofnonlinear mathematical models in ecology, namely, thatthe fluctuation patterns of abundances in many popula-tion systems can be explained, to a large extent, by nonlin-ear effects as predicted by simple (low dimensional) math-ematical models.

Despite the fact that mathematical and theoretical ecol-ogy developed and expanded profusely during the de-cades following May’s seminal work, his hypothesis hasproved both controversial and elusive to test. Efforts con-centrated on finding “chaos” in available historical datasets taken from field observations of animal populations.However, many formidable difficulties need to be over-come in order to provide a convincing argument for thepresence of chaotic dynamics in a biological population.These difficulties include the identification of the appro-priate state variables (phase space), the paucity of suffi-ciently long time series of data, missing data, lack of repli-cated data sets, and the ubiquitous presence of noise inecological data sets. Another major shortcoming is the un-

availability of mechanistically basedmodels that are tied rigorously to data,that is to say, models that can be pa-rameterized, statistically validated, andshown to provide quantitatively accu-rate descriptions and predictions of apopulation’s dynamics.

Nonetheless, there were concerted efforts– particularly during the late 1980s andthe 1990s – to find evidence of chaos inavailable ecological data sets. One ap-proach, taken by Professor W.M. Schaffer(of the University’s Ecology and Evolu-tionary Biology Department) and his col-leagues in their influential studies ofcertain ecological and epidemiologicaldata sets, was based on the “reconstruc-tion” theorems of F. Takens. These theo-

dynamic: these knots evolve by rigid motion, translatingin space and sliding along themselves without any changein their topological type.

Many more solutions with increasingly complex geomet-ric and topological properties can be obtained and stud-ied, using a growing number of old and new techniques,and involving many diverse fields of Mathematics. Theyinclude Bäcklund transformations and Floquet theory,methods of Algebraic Geometry and Perturbation expan-sions, Knot Classification theory and Symmetry methods.With my colleague Tom Ivey at the College of Charleston,we continue our investigations of higher soliton solutionsof the Vortex Filament Equation and of related geometricevolution equations, with a keen interest in uncoveringthe deep connections between the infinite number of sym-metries and the amazing complexity of “solitary smokerings”.

Annalisa Calini is currently an Associate Professor withthe Department of Mathematics at the College of Charles-ton. She received her Ph.D. in Applied Mathematics in1994 at The University of Arizona.

J. M. Cushing, ProfessorDepartment of MathematicsInterdisciplinary Program in Applied Mathematics

central goal in popula-tion biology and ecology

is to understand temporal fluc-tuations in population abun-dance. Such fluctuations, how-ever, often appear erratic andrandom, with annual vari-ances spanning several ordersof magnitude. For example, arecent literature survey showedthat annual numbers of newadults can vary by factors of

over 30 in terrestrial vertebrates, 300 in plants, 500 in ma-rine invertebrates, 2200 in birds1 . In the early 1970’s LordRobert May put forth a bold new assertion concerning thepossible explanation of the perplexing dynamic patternsso often observed in biological populations. The prevail-ing point of view had been that complex patterns havecomplex causes and simple causes have simple conse-quences. May’s theoretical work showed, on the otherhand, that complex patterns (including what is now called“chaos”) could result from simple rules.2 This non-intui-tive fact raised the intriguing possibility that some of thecomplexity of nature might arise from simple laws.

The complexity about which May wrote is a result ofnonlinearity. Although the classical mathematical mod-els of theoretical ecology from the first half of the twentiethcentury are nonlinear, the theories de-rived from them were centered on equi-librium dynamics. The famous logisticequation and the Lotka-Volterra systemsof competition and predation equationsare prototypical examples found in mostundergraduate textbooks on differentialequations and ecology. Fundamentally,the mindset at the heart of these theoriesencompassed the notion of a “balance ofnature” in which ecological systems areinherently at equilibrium and the erraticfluctuations and complexity observed indata are due to “random disturbances”(or “noise”). From this point of view, eco-systems are noisy perturbations of under-lying stable configurations. The point ofview suggested by May, however, was not

COMPLEX POPULATIONDYNAMICS

1 N. G. Hairston Jr., S. Ellner, and C.M. Kearns, Population Dynamics inEcological Space and Time (O. E.Rhodes, R. K. Chesser and M. H.Smith, eds.), University of ChicagoPress, 1996, 109-145

2 Lord R M. May was not the first tostudy and write about chaos, al-though he, together with E. Lorenz,did as much as anyone to stimu-late interest in complex dynamicsand “chaos”. This interest had laindormant in the scientific commu-nity since Henri Poincaré stumbledacross chaos nearly one hundredyears earlier.

3 R. M. May, Nature 261 (1976),459-467

REFERENCES

1. A. Calini, Recent developments in integrable curve dy-namics. Geometric Approaches to Differential Equations, Lec-ture Notes of the Australian Mathematical Society, 15 (2000),56-99.

2. A. Calini, T. Ivey, Connecting Geometry, Topology and Spec-tra for Finite-Gap NLS Potentials,Physica D, 152-153 (2001), 9-19.

3. G.K. Batchelor, An Introduction to Fluid Dynamics, Cam-bridge University Press (1967).

Figure 4. A gallery of solitary smoke rings. From left to right, top row: an unknot, a trefoil knot (a (2,3) torus knot), a (2,5) torus knot,bottom row: (2,9), (3,8) and (4,9) torus knots.

2 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 15

rems allow attractors to be studied from knowledge of onlya small number of a system’s state variables and therebyprovide a means to investigate chaos in a lower dimen-sional and more tractable setting. Another approach takenby other researchers was based on attempts to determinewhether any available time series of ecological data dem-onstrates a dynamic property characteristic of chaos called“sensitivity to initial conditions”

4. This was done by uti-

lizing statistical methods – based on fitting the data withelaborate (often parameter rich) phenomenological mod-els – for the ultimate purpose of estimating a single diag-nostic quantity, called the “dominant Lyapunov expo-nent”, whose positivity indicates this signature propertyof chaos. The insufficient length of available data setsproved to be a severe constraint on theseapproaches. Another significant diffi-culty is the presence of high levels ofnoise in ecological data. Given the simi-larity of chaos to random noise, howdoes one distinguish between the two?

Because of these difficulties it is perhapsnot surprising that results from this“hunt for chaos” were equivocal. Nodata sets were found to be convincinglychaotic, according to the tests used, al-though some were judged tantalizingly“on the edge of chaos.” By the end of thecentury, various opinions were formu-lated which ranged from “chaos is rarein nature” to “the jury is still out.” Fromthe first opinion arises the question “whyis chaos rare in nature?” especially inlight of the fact that ecological modelsabound with chaotic dynamics. With thelatter opinion, one recognizes that the for-midable difficulties involved in detect-ing chaos in ecological data have notbeen adequately overcome by the phe-nomenological methods of time seriesanalysis. Researchers have had to relyon such methods until adequate mecha-

nistic models and data are available. However, as J. N.Perry puts it in a recent book that surveys these issues,“the consensus [among population ecologists] is that thereis no substitute for a thorough understanding of the biol-ogy of the species, allied to mechanistic modeling of dy-namics using analytic models, with judicious cautionagainst over-parameterization.”5 .

The complexity of the natural systems, along with the in-herent difficulties in confidently linking data from suchsystems with theory and models, pointed to a need forcontrolled laboratory experiments–experiments designedand analyzed with the specific intent of testing the predic-tions of nonlinear population theory. Laboratory micro-cosms, while no substitute for field experiments, are one ofthe best ways to test theories and hypotheses. In a labora-tory setting one can carefully control environments andreduce or eliminate the effects of confounding elements,identify important and unimportant mechanisms, makeaccurate census counts and other measurements, replicateresults, and manipulate parameters. For nearly decade J.M. Cushing (professor of mathematics and member of theProgram on Applied Mathematics) has worked with aninterdisciplinary team of biologists, statisticians and math-ematicians6 (with the support of the National Science Foun-dation) on a variety of projects designed to investigate non-linear phenomena in a real biological population by meansof controlled laboratory experiments tied closely to a math-

ematical model and its predictions. Oneof the team’s major efforts involved an in-vestigation of May’s famous tenet con-cerning complex and chaotic dynamics.

May’s approach to chaos was to ask howa population’s dynamics change if someof its fundamental demographic charac-teristics change. He looked at equationsof the form xt+1 = f(xt) as mathematicalmodels for the prediction of populationabundance (or density) xt from one cen-sus time to the next. Given an initial popu-lation density x0 such an equation pre-dicts recursively a unique sequence of fu-ture population densities x0 , x1 , x2 , … .In specific cases the sequence dependson the numerical values assigned to “pa-rameters” appearing in the equation (i.e.,“coefficients” in the expression f(x) ). Fa-mous examples include the “discrete lo-gistic” equation that uses f(x) = b x (1 - cx)and the more biologically applicable“Ricker” equation with f(x) = b x e-cx . Maywas interested in how the sequence of pre-dictions changes if the number b changes.More precisely, he was interested in thelong term properties of the sequence–the“attractor”–which might be as simple as

which first exhibited “solitary waves” amongst its solu-tions. A “solitary wave,” or “soliton” (as it was later called),is a hump-shaped solution which moves without chang-ing its shape and interacts elastically with other like solu-tions. Solitons sparked and have continued to renew vividinterest amongst communities of physicists (searching forparticle-behaving waves) and applied mathematiciansmodeling anything from rogue waves in the North Sea tooptical signals in the Trans-Pacific cable.

As a Ph.D. student of Nick Ercolani at the University ofArizona, I became interested in an equation describingsmoke ring dynamics, which provides a remarkable andperhaps the richest application of soliton theory to therealm of curve geometry.

A ring of smoke travelling through the air (as those pro-duced by our grandfather’s pipes) is a region of vorticityconfined to a thin filament in space. Vortex filaments arecommonly observed in fluids and superfluids, in plasmas,and as far away as in the flares of the solar corona and instellar atmospheres as magnetic arches. They are often open(like pieces of string), but can be closed curves, like smokerings and plasma loops, and they can even be knotted.

As a first approximation, the evolution of a smoke ring issuprisingly simple, and yet rich in symmetries (there is aninfinite number of symmetries, in fact). By modeling a verythin vortex filament as an embedded curve C in 3-spaceparametrized by arclength s, we introduce at each pointalong the curve its unit tangent vector T, the unit normalvector N (directed like the radius of curvature) and com-plete the pair with the binormal vector B to obtain an or-thonormal triple (T, N, B) (the Frenet-Serret frame of thecurve).

Assuming the smoke ring moves without stretching underthe effect of its own vorticity, one derives the Vortex Fila-ment Equation1

where κ(s,t) is the curvature of the curve at the point r(s,t).The equation indicates that the higher the curvature, thefaster the motion in the direction of the binormal vector,just as the smaller the smoke ring, thefaster it travels upward.

This nonlinear partial differential equa-tion possesses a wealth of conservedquantities: the total length is con-stant in time; neither the total squaredcurvature, , nor the total torsion canchange during the dynamics, and the in-tegral of the torsion times the curvaturesquare will also remain invari-ant while the curve evolves, and so will

an infinite number of conserved quantities expressed interms of the curvature κ and torsion τ. This is the mainsignature of a soliton equation.2

What do solitary wave solutions of the Vortex FilamentEquation look like? The simplest one, the zero-soliton (withno humps) is readily discovered: one can verify that the

planar circle of radius and position vector r(s,t) =

is a solution which rig-

idly translates with speed k0 in the k direction.

One-soliton solutions are more complex, and far more in-teresting objects than translating circular smoke rings.One-humped solitons are typically obtained as travellingwave solutions of the nonlinear partial differential equa-tion, by assuming the position vector r to be a function ofthe single phase variable (c is the speed of thetravelling wave) and so reducing the problem to an ordi-nary differential equation. Just like for the Kepler model,the presence of symmetries allows one to find explicit for-mulas for such solutions, involving, in this case as well,elliptic functions.

The resulting curves are shown in Figure 4, they turn outto be beautifully symmetric knots of a spe-cial type, well known to topologists. Onecan imagine creating a knot by wrappinga piece of string on the surface of a do-nut: p times along the meridians and qtimes around its longitudes, and then glu-ing the ends together. After eating the do-nut, what is left is a (p,q) torus knot. Torusknots of every (p,q)-type are realized byone-soliton solutions of the Vortex Fila-ment Equation: a remarkable way inwhich symmetry manifests itself. Symme-tries are also responsible for a very simple

1 More realistic models of vortexfilament dynamics account forstretching, non—local vorticityeffects as well as for non—zerothickness of the filament core (adiscussion can be found in {3}

2 The curvature κ(s) of a param-etrized curve measures theacceleration of the position rectorr(s), while the torsion τ(s) mea-sures how rapidly the curvedeviates from its osculating plane-τ=0 for a planar curve.

4 This property means thatpopulations starting close to eachother rapidly (exponentially)diverge from each other with time,eventually to becomeuncorrelated.

5 J. N. Perry, R. H. Smith, I. P.Woiwod, and D. R. Morse, Chaosin Real Data: the Analysis ofNonlinear Dynamics from ShortEcological Time Series, KluwerAcademic Publishers, Dordrecht,The Netherlands, 2000

6 R. F. Costantino (Department ofBiological Sciences, University ofRhode Island), Brian Dennis(Department of Fish and WildlifeResources and Division ofStatistics, University of Idaho),Robert A. Desharnais (Depart-ment of Biology and Microbiol-ogy, California State University),Shandelle M. Henson (Depart-ment of Mathematics, AndrewsUniversity), and Aaron A. King(Department of EnvironmentalScience and Policy, University ofCalifornia at Davis). Supportedby the National Science Founda-tion.

The Beetle Team: Shandelle Henson, Jim Cushing, RobertDesharnais, Brian Dennis, and Bob Costantino

Figure 3. A space curve of position vector r and Frenet frame(T, N, B)

14 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 3

an equilibrium point (the sequence’s limit point, if itshould happen to converge) or a periodic cycle (a 2-cycle,for example, in which the sequence oscillates alternatelybetween two different numbers.) Or, as May discovered,the attractor might be a very complicated oscillation in-distinguishable from random noise. Which attractor re-sults, depends on which value of b is used in the formulaf(x). Changes in the type of attractor caused by changes inb are called “bifurcations” and the graphical summary ofthese bifurcations depicted in Figure 1 for the Ricker equa-tion has become a virtual icon of chaos theory. It shows aprogressive complexity in attractor dynamics as b is in-creased, a “route-to-chaos.”

Will a real biological population exhibit a sequence of dy-namic bifurcations, and a route-to-chaos, like those pre-dicted by such simple mathematical equations? There is ademanding prerequisite required for answering this ques-

tion, namely, the identification of an adequate mathemati-cal model for the particular population involved; a modelthat provides more than good statistical fits to data. Themodel needs to be biologically-based, that is to say, de-rived from biological mechanisms known to be importantfor the dynamics of the population under consideration,and it needs to be quantitatively accurate in its predic-tions. Unfortunately models that “work” in this sense arerare in population dynamics and ecology. The track recordof relating mathematical models to data is not good, par-ticularly when it comes to the successful formulation ofhypotheses and predictions that are quantitatively test-able. Therefore, a major hurdle faced by Cushing and hiscolleagues in conducting a study aimed at documenting aroute-to-chaos in a real biological population was the math-ematical modeling of the population dynamics of the or-ganism used in the study–the common flour beetle(Tribolium castaneum).

Cultures of flour beetles make anideal experimental system forsuch a study for several reasons.First of all, a great deal is knownabout their biology (genetics,physiology, and behavior). Thisis primarily because they are asignificant agricultural pest (thathas contaminated stored grainproducts since ancient Egyptiantimes) and therefore have been

the object of scientific scrutiny for many decades. Further-more, cultures are easy to maintain, easy to census accu-rately, and easy to manipulate. Long time series of censusdata can be gathered in a reasonable length of time sincethe beetle produces, under normal laboratory conditions,a new generation in only four weeks. Important for theproject is that they have a complete insect life cycle (lar-vae, pupae, and adult) with inter-stage interactions (can-nibalism) that induce nonlinear effects. These nonlineareffects, which biologically regulate population densityeven when a continual supply of food resources is avail-able, provide the necessary ingredients for dynamiccomplexity.

The mathematical model devised by the “Beetle Team” pre-dicted population numbers from one census time to thenext, in a manner similar to the famous Ricker equationexcept that each data point consists of three numbers in-stead of one, namely, counts of each life cycle stage. Themodel uses larval, pupa and adult stages as its state vari-ables so as to account for the dominant biological mecha-nism that drives the nonlinear dynamics of the beetle cul-tures — the cannibalism practiced by individuals in onestage on those of another. The “LPA model” is three di-mensional, unlike the one-dimensional Ricker model, andtherefore is more difficult to analyze mathematically. Whilesome mathematical results have been obtained, there re-

Annalisa Calini, Associate ProfessorDepartment of Mathematics, College of Charleston

he main successes in solving nonlinear differentialequations come when the equations have lots of sym-

metries, often appearing as conserved quantities, like en-ergy and momentum. One well-known example is theKepler problem for planetary motion around the sun, forwhich conservation of angular momentum allows one tosolve the equations explicitly.

When the equation contains partial derivatives, in certainlucky situations, there is an infinite number of conservedquantities, some of them carrying a direct physical inter-pretation, most coming from more abstract symmetries.Among these equations is the famous Korteweg-de Vries(KdV) equation, a model of water waves in shallow water,

The response to the AWS has been tremendous. Our origi-nal proposal called for inviting and funding the expensesof 30 graduate students each year. Last year, we stretchedour funds with matching contributions from various de-partments including Princeton, Harvard, and Stanford,and were able to fully or partially fund about 84 students. There were about 140 participants in all, so it’s clear thatmany people find the AWS a must-attend event, even ifthey have to spend their own money.

In addition to the AWS, the Southwestern Center grant (a“Group Infrastructure Grant” from the NSF) has spon-sored a distinguished lecture series, both at the UA andother participating institutions (which are the Universi-ties of Texas, New Mexico, and Southern California). AtArizona, we’ve had lecture series by Ken Ribet, JimCarlson, Anand Pillay, David Rohrlich, and AlexanderGoncharov. The Center also sponsors a post-doc (currentlyShuzo Takahashi), provides summer support and travelfor graduate students, and has bought computer equip-ment for the local principal investigators.

The local PIs on the grant are Minhyong Kim, BillMcCallum, Dinesh Thakur, and Doug Ulmer. The outsidePIs are Alex Buium at the University of New Mexico,Wayne Raskin at USC, and Felipe Voloch at UT Austin. We’ve benefitted from the collaboration of many othermathematicians, and the tremendous efforts of SandySutton who handles all our administrative work.

For more about the activities of the Southwestern Center,especially the upcoming Arizona Winter School (March9-13, 2002), see our web site athttp://swc.math.arizona.edu/

SOLITARY SMOKE RINGS

Figure 1. The equation for planetary motion

has the group of rotations as symmetries. Correspondingly,conservation of angular momentum L reduces the model to aplanar system of equations solvable by quadratures

FIGURE 1. The upper graph shows the attractor of theRicker equation plotted against the corresponding value of b(with c = 0.01). For small values of b, less than approxi-mately 7.4, there is a single point lying on the graph, indicat-ing that the attractor is an equilibrium point. For values of bbetween approximately 7.4 and 12.5 there are two points onthe graph, indicating the attractor is a periodic 2-cycle. Forincreasing values of b, the attractor consists of more andmore points (with some exceptions). For b = 30 the timeseries of x plotted against t in this case, shown in the lowergraph, exhibits the kind of random appearing oscillations towhich May refers in his seminal paper.

Hotel Work Session

4 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 13

Figure 2. Interaction of two solitons of the KdV equation.Frames are shown for four consecutive values of time

main many interesting unsolved problems associated withthe model. Nonetheless, the simple recursive nature of thethree equations in the model makes the model easy to studyusing computers.

In order to connect the LPA model to real beetle data it wasnecessary for the team to “parameterize” or “calibrate”the model. This means numerical estimates for the param-eters appearing in the model (of which there are six) neededto be obtained and the resulting fitted model shown to beaccurate. This statistical effort required a description ofthe inevitable deviations of real data from the model pre-dictions. In general such deviations, or “noise,” can arisefrom a variety of different causes. In the case of data fromthe laboratory cultures of flour beetle, however, there isvirtually no census error and, because of the highly con-trolled laboratory conditions, there is minimal “environ-mental noise,” i.e., few external random disturbances thatsignificantly affect the populations as a whole (such aschanges in temperature, humidity, light, or food resources).The source of the noise is “demographic,” that is to say,results from random differences among individuals in thepopulations–differences that are ignored by the model(gender, chronological age, body size, egg laying rates, mor-tality rates, etc.). The team had to devise mathematicalways to describe this precise kind of “stochasticity” withinthe beetle system and then incorporate demographic noiseinto a stochastic version of the LPA model. With this modelin hand, a large number of statistical techniques becameavailable for both the parameterization and the validationof the model.

In the mid 1990’s the team successfully demonstrated theaccuracy of the LPA model in preliminary tests that usedexisting historical data sets for flour beetle populations. Amore important test of the prediction capability of the modelfollowed when the team designed and successfully imple-mented a yearlong experiment that documented a predictedperiod doubling bifurcation. This bifurcation is like thatoccurring in the Ricker model except in this preliminaryexperiment the prediction did not include chaos. None-theless, this work was praised in an editorial appearing inthe journal Nature7 (where the experiment was announcedby the Beetle Team) as “an unusual blend of nonlineardynamics theory, statistics and experimentation” with re-sults that “are of uncommon clarity for ecology” in show-ing “how the marriage of nonlinear models and experi-ments can help” ecologists accomplish the task of under-standing and anticipating the consequences of environ-mental perturbations. In short, the Beetle Team had pro-vided an example in ecology “of a model that actually livesup to [its] promise.”

With the preliminary work of deriving,analyzing and calibrating a model fin-ished, the Beetle Team was set to designand implement an experiment to test a

model predicted route-to-chaos. Two years later the teamannounced the results in the journal Science. In that sameissue, an editorial remarked that the Beetle Team had pro-vided “the most convincing evidence to date of complexdynamics and chaos in a biological population.” and that“ecologists at last have a convincing example of chaosthat they can use as a base to understand better complexdynamics in other laboratory systems and, more impor-tantly, in the field.”8 Furthermore, this project demonstratedquite clearly that a real biological population can indeedtraverse a route-to-chaos as predicted by a “simple” (i.e.,low dimensional) mathematical model, providing a directverification of May’s famous tenet.

While the demonstration ofchaos in a biological populationwas a milestone for the BeetleTeam, it was in many ways justthe beginning of its study of non-linear population dynamics. Theflour beetle system, with its vali-dated mathematical model and itsstatistical and laboratory compo-nents, constitute a powerful toolwith which to delve into the fas-cinating intricacies of nonlinear

theory and to discover new phenomena that are more thanjust mathematical theories.

The “chaotic” beetle populations from the route-to-chaosexperiment are still being maintained. Data from thesepopulations is now nearly seven years–nearly ninety gen-erations–in length. This long time series of data permitsthe Beetle Team to carry out a detailed study of a biologicalpopulation exhibiting chaotic dynamics. Chaotic dynam-ics, while appearing random in their oscillations, none-theless have discernible temporal patterns. They also paintdistinctive geometric patterns in state space. It is ratheramazing that the Team has been able to document the oc-currence of many such patterns predicted by the LPAmodel–subtle patterns involving not only “attractors”(around which most ecological theory revolves) but un-stable entities (called “saddles”). Furthermore, in a studyto appear in the journal Science later this year and led bythe newest Team member, Professor Shandelle Henson(former Hanno Rund Visiting Professor in the Departmentof Mathematics), unusual patterns in the data are shownto be “lattice effects,” i.e., caused by the fact that animalscome in whole numbers (mathematical models, like theLPA model, deal with mean numbers). This is yet anothersource of potentially predictable patterns in ecological data

whose occurrence was observed for thefirst time in real data by the Team.

These, and other studies, support a gen-eral tenet that has evolved from theTeam’s research. It is unlikely that one

Motivation is as much the responsibility of the student asthe professor.

In addition, Lesch and I began looking at a problem thatFriedlander told me about before my trip. Let us considera domain in R2 which is invariant when acted upon by agroup. Suppose in addition that the action of this groupcommutes with the Laplacian, where we are consideringDirichlet boundary conditions. Then the eigenspaces ofthe Laplacian are representation spaces of this group. Onenow asks a question involving the multiplicities of theirreducible representations occurring in the eigenspaces.Results along these lines have been obtained for domainswith dihedral symmetries. Much of my time in Köln wasspent trying to understand these results and attemptingto understand the problems with using similar techniquesto get results for cyclic groups.

I also spent some time in Aachen, a town about 40 mileswest of Köln on the border of Germany, the Netherlands,and Belgium where Tom Hoffman, another Arizona gradu-ate student studying representation theory of finite groups,was visiting the Nordrhein-Westfalen Institute of Tech-nology (RWTH). Hoffman’s advisor and Arizona profes-sor, Klaus Lux, was also in Aachen and I spent some timegrilling them about representation theory while we hap-pily drank our fill (and sometimes a wee bit more) of Ger-man beer.

The end of my trip was spent in Edinburgh, Scotland, at aconference entitled “Progress in Partial Differential Equa-tions.” The conference took place in the historic RoyalSociety of Edinburgh and I got the chance to listen to manygood talks by people from all over Europe and the U.S.One of the highlights of the conference for me was a talkby Nikolai Nadirashvili from the University of Chicago.His talk was entitled “New Results and Open Problemsin Elliptic PDE,” in which he discussed the current stateof elliptic PDE, including the problem I had been workingon in Köln. In addition to the talks, the conference was achance to meet European mathematicians and form con-tacts. Saadia Fahki of the Université Paris XII graciouslyoffered to show my wife and me around Paris if we shouldever make it there.

In sum, my trip was an unforgettable experience. I sawparts of Europe that I had never seen, and even learned“ein bisschen Deutsch.” I met other mathematicians withwhom I sincerely hope I will maintain at least a friend-ship, if not a working relationship. As I look back uponeverything that I saw and did, I realize I have only begunto reap the benefits of my time abroad.

Douglas Ulmer, Associate ProfessorDepartment of Mathematics

hat brings one hundred or so top graduate stu-dents and a handful of world experts in number

theory to Tucson during the second week of March eachyear? It could be the beautiful Sonoran desert weatherthat time of year, but a more likely reason is the ArizonaWinter School!

The Arizona Winter School (AWS), which is one of themain activities of the Southwestern Center, is an annualmeeting for advanced graduate students and post-docs inarithmetical algebraic geometry (or, more briefly, modernnumber theory). It has quite a different flavor from a tradi-tional conference—rather than having a loosely relatedsequence of talks, with each speaker reporting on his orher recent work, the AWS features a tightly integrated setof courses by leading and emerging experts on a theme ofcurrent interest. The program has included such lumi-naries as Pierre Deligne, Barry Mazur, Karl Rubin, andPeter Sarnak, as well as local talent like Bill McCallumand Doug Ulmer.

A characteristic feature of the AWS is that great effort isexpended to make sure that there is significant interactionbetween participants at all levels. For example, eachspeaker is assigned a group of students who work togetheron a project related to that speaker's course, and the stu-dents report on their projects at the end of the School. There are evening working sessions in the hotel (often last-ing until midnight) and several social events. The Winterschools have led to collaborations between researchers andstudents at various universities and to a real sense of ca-maraderie among the next generation of number theorists.

Another unique aspect of the AWS is the “professionaldevelopment component.” This part of each AWS is meantto give students some training in an aspect of the profes-sion beyond mathematics. The professional developmentcomponents have ranged from a workshop on teaching,(including viewing a tape from the Third InternationalMathematics and Science Study -the famous TIMSS studycomparing math education in various countries- with apanel discussion), to sessions on mathematical writing,jobs in industry, and symbolic computation.

THE SOUTHWESTERNCENTER FORARITHMETICALALGEBRAIC GEOMETRY

8 C. Godfray and M. Hassell,Science 275 (1997), 323

7 Peter Kareiva, Nature 375, 1995,189

12 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 5

topology for advanced undergraduates. The latter wasdesigned to prepare undergraduates for a lecture series tobe given by Bruno Harris, who will be visiting Nankaithis fall. Dr. Harris’s visit grew out of a conference in Tianjinin memory of K. T. Chen and W. L. Chow; Dr. Harris will belecturing on Chen’s iterated integrals. I tried to get thestudents as quickly as possible to a basic understandingof differential forms and integration on manifolds so thatI could touch on the idea of differential forms and De Rhamcohomology on path spaces. It was a challenge but thestudents were very good, most of them having been ac-cepted to Nankai after winning a national mathcompetition.

Beyond teaching and research, we really came to enjoydaily life in Tianjin. We lived on campus (as do most fac-ulty there); we would begin most days with an early walkaround campus and then get breakfast at a local outdoormarket. We did essentially all of our shopping at open-airmarkets, which were much more convenient and acces-sible than your average grocery store in the U.S. Every-thing we ever needed was within an easy (and safe) walkor bike ride from home.

Food was a highlight of our time in China, and it willprobably be a while till we dare to go to a Chinese restau-rant in the U.S. Ultimately, though, it was the social atmo-sphere of our many wonderful Chinese meals that madethem so special. The best meals were at friends’ homesand most special of all were the meals that we were luckyenough to share with Professor Chern, a kind, culturedand very gracious host. Several times during the year wewere invited to dinner at his house and it was always agreat honor. When Chern turned 89 the Institute arrangeda festive birthday dinner at a fancy campus restaurantwith many friends and mathematicians from aroundTianjin and Beijing; the dishes were too numerous to keeptrack of but I do remember delicious crabs.Perhaps the most surprising thing about our visit was thatit was so novel. Chinese mathematicians regularly experi-ence the mathematical environment in the U.S. and bringback new perspectives to China, yet the situation is rarely

can provide adequate explanations of ecological data bymeans of model attractors alone (or even fuzzy stochasticversions of attractors). Instead, one must expect to observepatterns attributable to a variety of deterministic entities,both stable and unstable and also transient and asymp-totic, all mixed into randomly occurring episodes bystochasticity. Nonetheless, what the Team has also shown–in at least one laboratory system–is that this rather daunt-ing mix of complexity can be sorted out by means of a lowdimensional mathematical model.

The Beetle Team’s decade long project on complex dynam-ics and chaos in population dynamics is the subject of theinaugural book in a new series on Theoretical Ecology,scheduled to be published by Academic Press next sum-mer. The Beetle Team has also carried out other experi-ments designed to delve further into the complexities ofchaos. For example, an experimental documentation of adynamic property called “sensitivity to initial condi-tions”–widely recognized as the signature characteristicof chaotic systems–was recently published by the Team inthe journal Ecology Letters. In another example the Team,together with Dr. Aaron King (a former student of W.M.Schaffer and also a Flinn Postdoctoral Fellow in the Ap-plied Mathematics Program at the University of Arizona),has recently completed new experiments designed to studysublte model predicted, temporal patterns within the cha-otic dynamics of the beetle cultures (patterns involvingcomplicated quasi-periodic motion with various rotationnumbers associated with unstable cycles embeddedwithin the chaotic attractor).

The Beetle Team, with its unique interdisciplinary blendof mathematics, statistics, and experimentation, has notrestricted its studies in nonlinear dynamics to chaotic dy-namics. Past studies have included investigations of mul-tiple attractors, resonance in periodically fluctuating habi-tats, phase shifting, and saddles and their stable mani-folds. Characteristic of these studies is that they have, insome cases, provided explanations previously unavail-able for patterns that had been observed in data and, inother cases, the identification and documentation of un-expected and non-intuitive phenomena predicted by theLPA model.

Currently studies are underway that extend the Team’sresearch in several new directions. For example, a modifi-cation of the model has been formulated to include thegenetics involved in flour beetle adaptation to the insecti-cide Malathion. An experiment is under way that tests theprediction of this model that the beetle dynamics will un-dergo a dynamic bifurcation as the population geneticallyadapts to the insecticide. Another example involves a mul-tiple species version of the LPA model studied by Dr. JeffEdmunds, recent graduate of the Department of Mathemat-ics. In his thesis, Dr. Edmunds showed that two compet-ing flour beetle species, placed in the same habitat, do not

always support a classical tenet of ecology concerningcompetitive exclusion–a principle on which the notion ofecological niche is based. This is particularly interestingbecause flour beetles were used in early studies that sup-ported that now widely accepted theory. However, therewere anomalies in those early experiments that Dr.Edmunds hopes to explain and with this as a startingpoint the Team hopes to conduct experiments that lead tonew insights into competition theory.

The interdisciplinary projects carried out by the BeetleTeam have brought mathematical models into a closer con-nection with population biology and ecology. One of theTeam’s hopes is that its successes not only provide in-sights into nonlinear population dynamics, but that theyprovide a modest step towards the “hardening” of eco-logical science–a step towards raising its explanatorypower beyond purely theoretical speculation and a satis-faction with only qualitative accuracy, reasonable “guess-timates,” and verbal metaphors. It is true that the labora-tory system used in the flour beetle experiments is a rela-tively simple biological system, and that the low dimen-sional LPA model is a simple mathematical model of thatsystem. Nonetheless, we can find motivation and inspira-tion for the study of simple systems from May’s seminal1976 paper:

“Not only in research, but also in the everyday worldof politics and economics, we would all be better offif more people realized that simple nonlinear sys-tems do not necessarily possess simple dynamicproperties."

Fred Stevenson, ProfessorDepartment of Mathematics

he University of Arizona is establishing a Center forRecruitment and Retention of Secondary Mathemat-

ics Teachers in the Department of Mathematics. The Cen-ter is in response to a critical shortage of qualified second-ary mathematics teachers. This crisis is national in scope;experienced teachers are retiring or quitting in droves; itis estimated that 35% of new mathematics teachers leavethe profession within five years and 30% of Universitystudents who get a teaching degree in mathematics do not

reversed. It seems that foreign mathematicians have vis-ited for a few weeks or a few months, but hardly any seemto have visited for a whole year or more.

I’m sure it will only get easier in the future to arrange suchvisits; Chinese mathematicians want more visitors andoften asked for my ideas on how to attract them. My posi-tion was not advertised and didn’t even exist until I askedabout it, but all it took was the initiative to ask and a goodword from my advisor. A good place to start asking mightbe the mathematical division of the Chinese Academy ofSciences in Beijing, since they do have a lot of short-termvisitors at the moment. Even without any Chinese lan-guage, life in a major Chinese city is quite easy to adapt to,and one could always ask the host institution to organizea crash course in basic Chinese. I am certain that anyfuture mathematical visitors to China will encounter thesame warm-heartedness, generosity and hospitality in hisor her daily interactions that made our visit so easy and somemorable.

Jeff Selden, Graduate StudentDepartment of Mathematics

spent the month of June working with Matthias Leschat the University of Köln in Germany. Lesch, who stud-

ies global analysis and geometry, was an Associate Pro-fessor in the Department at the University of Arizona untilJanuary, 2001, when he took a position at the University ofKöln. I began working with him in the spring of 2000when I was preparing for the oral preliminary examina-tion, which entails studying and then presenting a talk ona current research paper. After he left, I started workingwith Lennie Friedlander, a Professor in the Department,who studies spectral theory and geometry. In order to keepa working relationship with Lesch, I applied for and re-ceived a VIGRE grant which covered the airfare for my tripto Europe.

Upon arrival in Germany, I met Lesch’s doctoral student,Christian Frey. Together we studied K-theory for C*-alge-bras and I gave two introductory talks on the subject to agroup of pre-diploma students in Köln. Working withChristian in preparing my talks was an invaluable experi-ence; not only for the mathematical benefits of collabora-tion but for the education I received in the German studentattitude. German students do not have to pay to go touniversity. While they are there, no one forces them to dohomework, attend class or study. It is up to the individualstudent to decide why they want to study a certain subject.

CENTER FORRECRUITMENT &RETENTION OFSECONDARY MATHTEACHERS

A GRADUATE EXPERIENCEIN GERMANY

6 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 11

enter the classroom at all, but opt instead for the highersalaries and prestige of other professions. We mirror thesituation here in Tucson. Ten years ago the University ofArizona would provide the school system with at least 15new mathematics teachers per year; this year we have onlytwo on track to graduate.

The shortage of qualified mathematics teachers has beenexacerbated by a recent mandate from the State Board ofEducation. Last year the Board ordered that all enteringhigh school students must take two consecutive years ofmathematics covering material that has traditionally beentaken only by the better math students. In most schoolsthis meant that all students were to take Algebra 1 theirfreshmen year and Geometry their sophomore year. Unfor-tunately middle schools cannot prepare all students forAlgebra 1 in the 9th grade. The failure rate for freshmenwas expected to be bad and it turned out to be embarrass-ing: in many high schools more than half the studentsfailed freshman mathematics. This year, those failed stu-dents are in 10th grade Geometry, and failure rates mayvery well be even higher. The mandate was made neces-sary because of the creation of the AIMS test (Arizona In-strument to Measure Standards), the Arizona state highschool exit examination. This test is first given to all stu-dents at the end of their sophomore year and, on the ad-vice of lawyers, it was decided that all mathematics testedby AIMS must be presented to the students before they takethe test. The test includes Algebra 1 and Geometry. Natu-rally schools are scrambling to salvage students by offer-ing supplementary remedial courses but finding qualifiedteachers is a virtual impossibility nowadays.

The Center is attempting to help with this difficult reality.Our first job is to address this emergency in the secondarymathematics classrooms. This fall we are piloting a pro-gram employing 20 tutors from the University and PimaCollege and 15 cooperating teachers from two middleschools and six high schools. The tutors are taking a onecredit hour course and receiving $10 per hour for theirtutoring. This is only a beginning; bear in mind that thereare close to 60 secondary schools in Tucson and several ofthem have at least 100 students who need and want extrahelp in mathematics. In the spring semester we begin asecond program placing as many as 40 high school stu-dents in middle schools and high schools. In the secondyear we hope to expand our programs even further, per-haps involving as many as 80 tutors and 40 coordinatingteachers. This is not only an initiative to help studentsand teachers in the schools but it is also a recruiting tool.There are many university students who have an aptitudefor mathematics and who love to help others. We want togive them the opportunity to experience what it is like touse their talents in a meaningful way.

We also plan a large scholarship initiative to attract youngstudents into teaching. Already this fall we have securedfunding to provide 13 scholarships to juniors and seniorsat the University who are considering a teaching career.In the future we hope to have as many as 25 scholarshipsand include high school seniors and university freshmenand sophomores as recipients.

In the near future we intend to initiate a program wherenew teachers and experienced teachers can link together,get to know each other, and provide mutual support, en-couragement, and advice. We hope that this will alleviatethe exodus from the classroom by first and second yearteachers. We also want to provide a professional compo-nent to the life of the classroom teacher. This involvesfunding teachers who wish to return to the classroom andtake advantage of our graduate courses in MathematicsEducation and also provide support for those who wishto attend and participate in regional and national profes-sional meetings.

This is an expensive undertaking. Setting up the Centerrequires finding space, hiring directors and staff, and pro-viding for operating funds. Carrying out our plans iscostly; in fact, elementary mathematics will show that thetutoring initiative alone has a surprisingly high price tag.We have begun operations thanks to generous gifts fromprivate individuals. These funds, along with some statefunds, have given the Center the chance to carry out itsplans for the first year and even begin outlining plans forthe second year. We have hired two of Tucson’s finestteachers to direct operations of the Center. Sue Adamshas been a mathematics educator for over 30 years in theTucson area. During the past two decades she has servedas Head of the Mathematics Department at University HighSchool and then as High School Mathematics Coordina-tor for Tucson Unified School District. Ann Modica hasalso been a mathematics educator in Tucson for over 30years. She has won a number of prestigious teachingawards including the Presidential Award for Excellencein Science and Mathematics Teaching. For the past de-cade she served as Head of the Mathematics Departmentat Rincon High School. And thanks go to Professor SteveWilloughby of the Mathematics Department who has pro-vided a temporary home for the Center in his office. Thisis an especially generous offer in our space-starved Math-ematics Building.

The Center has the support of the University Administra-tion. While it is not yet incorporated into the Universitystructure we have been given assurances that if the Centercan make a tangible difference in the next few years theUniversity will provide necessary resources and fund itlong term. We are hopeful that the work of the Center willmake a clear difference in the near future so that we cancontinue this work.

lating intellectually and exciting culturally. My wife and Imade close friends, enjoyed the small pleasures of dailylife in a Chinese city, ate incredible food, and accumu-lated quite a lot of frequent train rider miles travellingaround China.

During the last year of my doctoral program, while send-ing off a huge stack of job applications to universities allacross North America, I wondered what options were outthere to do mathematics elsewhere. I asked Alan Weinsteinat Berkeley for his thoughts on this. First he asked how myFrench was, and when I replied that it was pretty rustybut that I had studied some Chinese in college, he imme-diately said something to the effect of, “Well, just askChern, he can probably send you to China.”

Shiing-Shen Chern is a Chinese mathematician, an emeri-tus professor at Berkeley and one of the great modern dif-ferential geometers, so with some trepidation I sent him aletter asking how one might arrange a mathematical visitto China. First he called my advisor for a backgroundcheck, and then he called me and said, “So, you want togo to China? You should go to my institute.” Chernfounded the Nankai Institute of Mathematics in 1985, af-ter China opened up to the west and Chern began visitingChina again after so many years away. Now, at the age of89, he lives nearly full time in Tianjin, on the Nankai Uni-versity campus, and he is tremendously respected through-out China. It is not uncommon to meet taxi drivers in Chinawho know his name. Therefore, it is not surprising thatonce Chern took to the idea of my visit to Nankai, the restof the arrangements went very smoothly.

When I was offered a three year postdoctoral position atthe University of Arizona, the hiring committee was kindenough to let me take one year off to spend in China. Fur-thermore, my wife was kind enough to agree to come withme, and so all things conspired to make the trip a reality.After a first year and a summer in Tucson, we set off forChina in September 2000.

My work is in low-dimensional symplectic and contacttopology; while no one at Nankai works in precisely thisfield, there were many researchers and graduate studentseager to learn what they could about the subject and toteach me about their own related work in geometry, topol-ogy and analysis.

The Institute has a well-endowed mathematics library, sothat access to books and journals was never a problem.Early on in our visit a colleague, Fuquan Fang, and I orga-nized a working seminar on the topology of contact struc-tures and foliations with several graduate students and afew other researchers. Almost all mathematical discus-sions happened in English, since my Chinese is not goodenough for mathematics. The Institute had weekly collo-quia with speakers from all over China and often from

other countries; the theme was usually differential geom-etry and Chern was always an active participant unlesshe was out of town.

I often visited Peking University, an hour and a half trainride away, to give or listen to talks, and I attended severalinternational conferences around China. The most memo-rable was a symplectic geometry conference that spannedthree cities: we started in Chengdu, the capital of Sichuanprovince, then moved to Tianjin and finally ended up inBeijing. I gave a talk on my own work and learned a lotfrom an excellent mix of mathematicians from all acrossChina together with prominent symplectic geometers andtopologists from Japan, the U.S. and Europe.

The mathematical content was stimulating enough but justin case we needed more stimulation, we were served threebanquet-style meals each day; in Chengdu the food wasthe best, Sichuan being particularly well known for itscuisine. At our final lunch in Chengdu, a meal of Sichuan-style dimsum, the tables had motorized lazy susans thatseemed to magically sprout new dishes every time youlooked away. It was a tough meal to keep up with, andwhen we got back to Tianjin we skipped several meals justto rest our stomachs.

I taught two classes each semester, all quite a change frommy teaching routine in Tucson. The first semester I taughtbasic graduate differential topology (with six students)and a special “Mathematical English” course (with 30 stu-dents). Differential topology was lots of fun and we be-came good friends with all of the students. The Englishcourse was more difficult; I didn’t really feel qualified butobviously none of the real English teachers on campuswere qualified either. I decided to focus on oral skills, sincethe students seemed pretty good at reading and writing.We ended up swapping short lectures on elementary top-ics, so that they could all practice listening to lectures inEnglish as well as giving presentations.

The second semester I taught “Topics in low-dimensionalgeometric topology” and a special course on differential

10 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 7

Elias Toubassi, Professor,Department of MathematicsAssociate Head for the Entry Level Program

he Mathematics Department, in cooperation withlocal school districts, has developed a two-year

project to work with high school mathematics teachers todevelop curriculum material that helps mathematics teach-ers assist their students to meet the State standards asmeasured on the AIMS test. This collaboration has re-sulted in an Eisenhower grant proposal which wasfunded by the Arizona Board of Regents. The first work-shop was held in June 2000 and the second in June 2001.The participating districts are: Amphitheater, CatalinaFoothills, Flowing Wells, Marana, Sunnyside, and Tuc-son Unified School District.

The 2001 workshop was held from June 4 through June 15on the campus of the University of Arizona and:

•Provided a forum for teachers to create curriculummaterial that addresses the State standards in math-ematics;•Included instruction on mathematics topics such asprobability, statistics, and logic, that give teachers adeeper background in standards-related topics; and,•Focused some sessions to develop new ideas to teachtraditional subjects such as algebra and geometry.

In addition, participating teachers took part in two fulldays of in-services during the 2001-2002 academic year.The Fall in-service was held at Sahuaro High School inOctober and the Spring in-service will be held at FlowingWells High School in March, 2002. The highlight of thein-services is time when teachers share how the new cur-riculum lessons are being received by students in theirclassrooms. These programs result in the formation ofcommunity support among faculty and high schoolteachers.

Participating teachers received a $400 stipend, a smallcurriculum allowance, and a classroom set of graphingcalculators. The following faculty assisted with the work-shop: Christopher Goff, Brigitte Lahme, Carl Lienert, BinLu, Jerry Morris, Diann Porter, Maria Robinson, JosephWatkins, and Stan Yoshinobu.

David Lovelock, ProfessorDepartment of Mathematics

n Fall 1990 I was teaching honors Calculus 1 in ourmathematics computer classroom. We had just opened

this room – the first in the U.S. – and had made it acces-sible to students in wheelchairs, even though in my previ-ous 30 years of teaching, I had never had a student with aphysical disability in any of my classes. Imagine my sur-prise when one of the first students to enter the room wasin a wheelchair. This was John Olsen, a freshman in theDepartment of Electrical and Computer Engineering. Somewill remember him tearing around campus with a MickeyMouse flag attached to the top of an antenna fixed to hiselectric wheelchair.

John and I remained in touch after he finished Calculus 1.John graduated and started postgraduate studies in ECE.Then one day in 1997 he introduced me to another gradu-ate student, Ali Mehrabian, in Civil Engineering. Ali alsowhips around campus in an electric wheelchair. The twoof them then told me a story.

Ali and John had been remorsing over how few studentswith physical disabilities were studying Science, Engi-neering, Mathematics, or Technology (now known asSMET) at UA. They had done a web search, and stumbledupon a National Science Foundation (NSF) request for pro-posals to encourage more students to study SMET. Theywanted to write such a proposal, but didn’t know what to

do. They turned to Georgia Ehlers, the Coordinator ofGrant and Scholarship Development in the UA GraduateCollege. She advised them to find a faculty member tospearhead the project.

To cut a very long story short – and despite all my kickingand screaming explaining that I knew nothing about dis-ability issues – I became that person, and so Program AC-CESS (Accessing Career Choices in Engineering and theScienceS) was born, and my life was changed forever.

With the help of an interdisciplinary cross-campus group,Ali, Georgia, John, and I wrote the proposal, and the NSFfunded the $500,000 three-year request in July 1998. It is amulti-faceted project aimed at students with physical dis-abilities from middle school to graduate school.

We have undergrads mentoring school kids, and gradu-ates mentoring undergrads –all with physical disabilities.We give encouragement grants to school teachers to helpthem make their classes more accessible. Ali and Johngive in-class demonstrations at schools to show that theirhandicaps are not handicaps. We work with the Collegeof Architecture and have their sophomores analyze vari-ous labs on campus and suggest ways to make them moreaccessible. And we offer a summer camp (Camp ACCESS)for 12 middle school kids.

The camp is our pride and joy. It lasts for 2 ½ weeks, and isrun by Faith Bridges, an adjunct faculty member in themathematics department, who has experience as a middle/high school teacher. The camp emphasis is on science, noton disabilities. The program is a smorgasbord of hands-on activities and is presented by UA faculty from manydisciplines. These activities range from Grocery StoreBotany to Mammals With Wings, from Being a Paleontolo-gist to How Spaceflight Affects Humans, from PracticalDendrochronology to Experiments With Liquid Nitrogen,from Hands-On Math to Blood and Guts!, from Buildingand Testing Model Bridges to What’s Inside My Computer?The campers visit a clean room, play with an electron mi-

croscope, take a tour of IBM and spend time selecting sci-ence books at Borders bookstore.

The schedule for the Camp ACCESS 2001 can be seen at http://www.math.arizona.edu/~dsl/camp2001.htm.Photos are visible at http://w3.arizona.edu/~access/.

I have been involved with a number of rewarding andworthwhile activities in my 40-year career as a faculty mem-ber at universities around the world, but this has been themost rewarding and the most worthwhile. I can do nobetter than quote from a letter from one of the camper’sparents, the full text of which can be read at http://w3.arizona.edu/~access/letters/letters.html. (You mayneed a Kleenex while reading the full letter. I do.)

“I have thought a lot about what it is that madeyour camp so wonderful for my son Justin as wellas his fellow campers. To say you changed my son’slife is not hyperbole. When I brought Justin to camp2 weeks ago, I entered with a disabled child. Twoweeks later, I am going home with an able childwho happens to have challenges in life. He has gonefrom withdrawn and frustrated to friendly withother children and excited about learning. You havetaught him more in 2 weeks than you will everknow.”

By being frugal, and accepting donations, we will be ableto run Program ACCESS for an extra year beyond the endof the initial three, until summer 2002. After that, the fu-ture is unclear, but of all things we have done, we will tryto keep the camp going. Finally, I should report that Johnnow has his Ph.D. and is working for IBM. Ali is closebehind.

Thank you John and Ali for twisting my arm.

David T. Gay, Visiting Instructor, Postdoctoral ScholarDepartment of Mathematics

just returned from a year of teaching and research atNankai University and the Nankai Institute of Math-

ematics in Tianjin, China; this article is a report on mymathematical and cultural experiences there. For a goodnumber of mathematicians in the U.S. (those originallyfrom China) this story won’t have too many surprises butit might be entertaining to hear it told from the point ofview of someone not Chinese. To those who have neverbeen to China or even thought of visiting, I hope the storyis interesting and encourages more foreign mathemati-cians to visit China in an academic capacity.It was a wonderful year: productive mathematically, stimu-

A MYOPIC VIEW OFPROGRAM ACCESSOR HOW SOME STUDENTSCHANGED MY LIFE

Grocery store botany

ENRICHING HIGHSCHOOLMATHEMATICS

Building bridges - with toothpicks and jellybeans

8 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 9

A YEAR IN CHINA