Presentations, Symposia, Workshops A VIEW FROM THE CHAIR Nicholas Ercolani Nicholas Ercolani, Department Head and Professor, is the Professor Moshe Shaked has been invited to give a presenta- Department Head of Mathematics and Professor co-organizer of a Symposium on “Waves: Patterns and tion for the Third International Conference on Mathemati- In this Issue Turbulence” for the Annual Meeting of the American cal Methods in Reliability, Methodology, and Practice, June Association for the Advancement of Science in Boston, 17-20, 2002 at the Norwegian University of Science and ur Mathematics Department has traditionally embraced February 14-19, 2002. Technology in Trondheim. PAGE TWO a broad view of its mission. Faculty members, as well as Complex Population students, participate in a wide range of activities that extend Dr. Ercolani will be a Plenary Speaker for the AMS Spring Doug Ulmer, Associate Professor, is co-organizing the AMS Dynamics outside of the Department per se. These include contributions to Jim Cushing Eastern Section Meeting in Montreal May 3-5, 2002. & Italian Mathematical Union, Special Session on Arith- the profession of Mathematics at all levels, interdisciplinary re- metical Algebraic Geometry, Pisa, Italy June 12-16, 2002. search and service to the community. The present newsletter gives PAGE SIX Professor Hermann Flaschka will speak at the workshop on us the opportunity to describe recent developments in some of Center for Recruitment & Geometry, Mechanics, and Dynamics, for the Field Institute these endeavors. Professor Maciej Wojtjiwski will be an invited Speaker for Retention of Secondary in Toronto, August 7-11, 2002. the International Congress of Mathematics 2002 in Beijing, Math Teachers Many of our faculty participate in cross-disciplinary research China, August 20-28, 2002. Fred Stevenson Yi Hu, Associate Professor, is an invited speaker at the ICM and educational projects through the Department’s unique rela- 2002 Satellite Conference in Algebraic Geometry in Shang- PAGE EIGHT tion with the Interdisciplinary Program in Applied Mathemat- Nicholas Ercolani hai, China, August 13-17, 2002. Enriching High School ics. This issue of the newsletter illustrates two such connections. Mathematics One is the article by Jim Cushing that describes the interplay between modern Dynamical Elias Toubassi Systems theory and the modeling of complex biological and ecological systems. Another is the article by former Applied Mathematics graduate student Annalisa Calini whose research PAGE EIGHT applies techniques of nonlinear science to the study of classical problems in Differential A Myopic View of Geometry. Annalisa is one example of a number of former graduate students in both the Program ACCESS Mathematics and Applied Mathematics graduate programs whose advisors were Depart- Mathematics at UA David Lovelock mental faculty and who went on to successful research careers at the interface between pure NON-PROFIT ORG. and applied Mathematics. Department of Mathematics U.S. POSTAGE The Universtiy of Arizona PAID PAGE NINE PO Box 210089 TUCSON, ARIZONA One Year in China Funding from VIGRE and GIG grants as well as other sources support a number of ongoing Tucson AZ 85721-0089 PERMIT NO. 190 David T. Gay activities that enrich the research experiences of our students and faculty. Examples of this are described in the articles of Jeff Selden, David T. Gay and Doug Ulmer. PAGE ELEVEN A Graduate Experience Departmental faculty continue to be successful in garnering gifts and awards which enable in Germany us to maintain a multifarious array of outreach activities such as enrichment and support of Jeff Selden regional secondary Mathematics teachers, as described in the articles by Fred Stevenson and Elias Toubassi, and Program ACCESS whose story is recounted in David Lovelock’s article. PAGE TWELVE The Southwestern Center In future issues we will be telling you more about our programs and activities. To our for Arithmetical alumni, I would also like to extend an invitation to write to us. We would be delighted to Algebraic Geometry Douglas Ulmer hear from you and perhaps some of you would even like to share your experiences in these pages. So please contact us and let us know what you are doing! PAGE THIRTEEN Solitary Smoke Rings Contact us at: http://www.math.arizona.edu/~mcenter/alum/alum.html Annalisa Calini

PAGE SIXTEEN Presentations, Symposia, Workshops 2 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 15

based on “noisy equilibrium” states. As he put it in his COMPLEX POPULATION seminal 1976 paper, the fact that a simple, deterministic DYNAMICS equation “can possess dynamical trajectories which look like J. M. Cushing, Professor some sort of random noise has disturbing practical Department of Mathematics implications. It means, for example, that apparently Interdisciplinary Program in Applied Mathematics erratic fluctuations in the census data for an ani- mal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rig- central goal in popula- idly deterministic population growth relation- tion biology and ecology ship.”3 is to understand temporal fluc- tuations in population abun- May’s tenet raised the possibility of new ways to under- dance. Such fluctuations, how- stand ecological systems. Although unexplained noise ever, often appear erratic and will always be present in ecological data, these insights random, with annual vari- provided a broad new hypothesis concerning the role of ances spanning several orders nonlinear mathematical models in ecology, namely, that of magnitude. For example, a the fluctuation patterns of abundances in many popula- recent literature survey showed tion systems can be explained, to a large extent, by nonlin- that annual numbers of new ear effects as predicted by simple (low dimensional) math- adults can vary by factors of ematical models. over 30 in terrestrial vertebrates, 300 in plants, 500 in ma- rine invertebrates, 2200 in birds1 . In the early 1970’s Lord Despite the fact that mathematical and theoretical ecol- 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, Robert May put forth a bold new assertion concerning the ogy developed and expanded profusely during the de- bottom row: (2,9), (3,8) and (4,9) torus knots. possible explanation of the perplexing dynamic patterns cades following May’s seminal work, his hypothesis has so often observed in biological populations. The prevail- proved both controversial and elusive to test. Efforts con- ing point of view had been that complex patterns have centrated on finding “chaos” in available historical data dynamic: these knots evolve by rigid motion, translating REFERENCES complex causes and simple causes have simple conse- sets taken from field observations of animal populations. in space and sliding along themselves without any change quences. May’s theoretical work showed, on the other However, many formidable difficulties need to be over- in their topological type. 1. A. Calini, Recent developments in integrable curve dy- hand, that complex patterns (including what is now called come in order to provide a convincing argument for the namics. Geometric Approaches to Differential Equations, Lec- “chaos”) could result from simple rules.2 This non-intui- presence of chaotic dynamics in a biological population. Many more solutions with increasingly complex geomet- ture Notes of the Australian Mathematical Society, 15 (2000), tive fact raised the intriguing possibility that some of the These difficulties include the identification of the appro- ric and topological properties can be obtained and stud- 56-99. complexity of nature might arise from simple laws. priate state variables (phase space), the paucity of suffi- ied, using a growing number of old and new techniques, ciently long time series of data, missing data, lack of repli- and involving many diverse fields of Mathematics. They 2. A. Calini, T. Ivey, Connecting Geometry, Topology and Spec- The complexity about which May wrote is a result of cated data sets, and the ubiquitous presence of noise in include Bäcklund transformations and Floquet theory, tra for Finite-Gap NLS Potentials, nonlinearity. Although the classical mathematical mod- ecological data sets. Another major shortcoming is the un- methods of Algebraic Geometry and Perturbation expan- Physica D, 152-153 (2001), 9-19. els of theoretical ecology from the first half of the twentieth availability of mechanistically based sions, Knot Classification theory and Symmetry methods. century are nonlinear, the theories de- models that are tied rigorously to data, With my colleague Tom Ivey at the College of Charleston, 3. G.K. Batchelor, An Introduction to Fluid Dynamics, Cam- rived from them were centered on equi- 1 N. G. Hairston Jr., S. Ellner, and C. that is to say, models that can be pa- we continue our investigations of higher soliton solutions bridge University Press (1967). librium dynamics. The famous logistic M. Kearns, Population Dynamics in rameterized, statistically validated, and of the Vortex Filament Equation and of related geometric equation and the Lotka-Volterra systems Ecological Space and Time (O. E. shown to provide quantitatively accu- evolution equations, with a keen interest in uncovering of competition and predation equations Rhodes, R. K. Chesser and M. H. Smith, eds.), University of Chicago rate descriptions and predictions of a the deep connections between the infinite number of sym- are prototypical examples found in most Press, 1996, 109-145 population’s dynamics. metries and the amazing complexity of “solitary smoke undergraduate textbooks on differential rings”. equations and ecology. Fundamentally, 2 Lord R M. May was not the first to Nonetheless, there were concerted efforts the mindset at the heart of these theories study and write about chaos, al- – particularly during the late 1980s and Annalisa Calini is currently an Associate Professor with encompassed the notion of a “balance of though he, together with E. Lorenz, the 1990s – to find evidence of chaos in the Department of Mathematics at the College of Charles- nature” in which ecological systems are did as much as anyone to stimu- late interest in complex dynamics available ecological data sets. One ap- ton. She received her Ph.D. in Applied Mathematics in inherently at equilibrium and the erratic and “chaos”. This interest had lain proach, taken by Professor W.M. Schaffer 1994 at The University of Arizona. fluctuations and complexity observed in dormant in the scientific commu- (of the University’s Ecology and Evolu- data are due to “random disturbances” nity since Henri Poincaré stumbled tionary Biology Department) and his col- (or “noise”). From this point of view, eco- across chaos nearly one hundred years earlier. leagues in their influential studies of systems are noisy perturbations of under- certain ecological and epidemiological lying stable configurations. The point of 3 R. M. May, Nature 261 (1976), data sets, was based on the “reconstruc- view suggested by May, however, was not 459-467 tion” theorems of F. Takens. These theo- 14 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 3

which first exhibited “solitary waves” amongst its solu- nistic models and data are available. However, as J. N. tions. A “solitary wave,” or “soliton” (as it was later called), Perry puts it in a recent book that surveys these issues, is a hump-shaped solution which moves without chang- “the consensus [among population ecologists] is that there ing its shape and interacts elastically with other like solu- is no substitute for a thorough understanding of the biol- tions. Solitons sparked and have continued to renew vivid ogy of the species, allied to mechanistic modeling of dy- interest amongst communities of physicists (searching for namics using analytic models, with judicious caution particle-behaving waves) and applied mathematicians against over-parameterization.”5 . modeling anything from rogue waves in the North Sea to optical signals in the Trans-Pacific cable. The complexity of the natural systems, along with the in- herent difficulties in confidently linking data from such As a Ph.D. student of Nick Ercolani at the University of systems with theory and models, pointed to a need for Arizona, I became interested in an equation describing controlled laboratory experiments–experiments designed smoke ring dynamics, which provides a remarkable and The Beetle Team: Shandelle Henson, Jim Cushing, Robert and analyzed with the specific intent of testing the predic- perhaps the richest application of soliton theory to the Desharnais, Brian Dennis, and Bob Costantino tions of nonlinear population theory. Laboratory micro- realm of curve geometry. cosms, while no substitute for field experiments, are one of rems allow attractors to be studied from knowledge of only Figure 3. A space curve of position vector r and Frenet frame the best ways to test theories and hypotheses. In a labora- (T, N, B) a small number of a system’s state variables and thereby A ring of smoke travelling through the air (as those pro- tory setting one can carefully control environments and provide a means to investigate chaos in a lower dimen- duced by our grandfather’s pipes) is a region of vorticity an infinite number of conserved quantities expressed in reduce or eliminate the effects of confounding elements, sional and more tractable setting. Another approach taken confined to a thin filament in space. Vortex filaments are terms of the curvature κ and torsion τ. This is the main identify important and unimportant mechanisms, make by other researchers was based on attempts to determine commonly observed in fluids and superfluids, in plasmas, signature of a soliton equation.2 accurate census counts and other measurements, replicate whether any available time series of ecological data dem- and as far away as in the flares of the solar corona and in results, and manipulate parameters. For nearly decade J. onstrates a dynamic property characteristic of chaos called stellar atmospheres as magnetic arches. They are often open What do solitary wave solutions of the Vortex Filament 4 M. Cushing (professor of mathematics and member of the “sensitivity to initial conditions” . This was done by uti- (like pieces of string), but can be closed curves, like smoke Equation look like? The simplest one, the zero-soliton (with Program on Applied Mathematics) has worked with an lizing statistical methods – based on fitting the data with rings and plasma loops, and they can even be knotted. no humps) is readily discovered: one can verify that the interdisciplinary team of biologists, statisticians and math- elaborate (often parameter rich) phenomenological mod- ematicians6 (with the support of the National Science Foun- planar circle of radius and position vector r(s,t) = els – for the ultimate purpose of estimating a single diag- As a first approximation, the evolution of a smoke ring is dation) on a variety of projects designed to investigate non- nostic quantity, called the “dominant Lyapunov expo- suprisingly simple, and yet rich in symmetries (there is an linear phenomena in a real biological population by means nent”, whose positivity indicates this signature property infinite number of symmetries, in fact). By modeling a very is a solution which rig- of controlled laboratory experiments tied closely to a math- of chaos. The insufficient length of available data sets thin vortex filament as an embedded curve C in 3-space ematical model and its predictions. One idly translates with speed k in the k direction. proved to be a severe constraint on these parametrized by arclength s, we introduce at each point 0 of the team’s major efforts involved an in- approaches. Another significant diffi- along the curve its unit tangent vector T, the unit normal 4 vestigation of May’s famous tenet con- One-soliton solutions are more complex, and far more in- culty is the presence of high levels of This property means that vector N (directed like the radius of curvature) and com- populations starting close to each teresting objects than translating circular smoke rings. cerning complex and chaotic dynamics. plete the pair with the binormal vector B to obtain an or- noise in ecological data. Given the simi- other rapidly (exponentially) One-humped solitons are typically obtained as travelling thonormal triple (T, N, B) (the Frenet-Serret frame of the larity of chaos to random noise, how diverge from each other with time, wave solutions of the nonlinear partial differential equa- eventually to become May’s approach to chaos was to ask how curve). does one distinguish between the two? tion, by assuming the position vector r to be a function of uncorrelated. a population’s dynamics change if some of its fundamental demographic charac- the single phase variable (c is the speed of the Because of these difficulties it is perhaps Assuming the smoke ring moves without stretching under 5 J. N. Perry, R. H. Smith, I. P. teristics change. He looked at equations travelling wave) and so reducing the problem to an ordi- not surprising that results from this the effect of its own vorticity, one derives the Vortex Fila- Woiwod, and D. R. Morse, Chaos of the form x = f(x ) as mathematical nary differential equation. Just like for the Kepler model, “hunt for chaos” were equivocal. No t+1 t ment Equation1 in Real Data: the Analysis of models for the prediction of population the presence of symmetries allows one to find explicit for- data sets were found to be convincingly Nonlinear Dynamics from Short abundance (or density) x from one cen- mulas for such solutions, involving, in this case as well, chaotic, according to the tests used, al- Ecological Time Series, Kluwer t Academic Publishers, Dordrecht, sus time to the next. Given an initial popu- elliptic functions. though some were judged tantalizingly where κ(s,t) is the curvature of the curve at the point r(s,t). The Netherlands, 2000 lation density x such an equation pre- “on the edge of chaos.” By the end of the 0 The equation indicates that the higher the curvature, the dicts recursively a unique sequence of fu- The resulting curves are shown in Figure 4, they turn out century, various opinions were formu- 6 faster the motion in the direction of the binormal vector, R. F. Costantino (Department of ture population densities x , x , x , … . to be beautifully symmetric knots of a spe- lated which ranged from “chaos is rare Biological Sciences, University of 0 1 2 just as the smaller the smoke ring, the In specific cases the sequence depends cial type, well known to topologists. One in nature” to “the jury is still out.” From Rhode Island), Brian Dennis faster it travels upward. 1 (Department of Fish and Wildlife on the numerical values assigned to “pa- More realistic models of vortex can imagine creating a knot by wrapping the first opinion arises the question “why filament dynamics account for Resources and Division of rameters” appearing in the equation (i.e., a piece of string on the surface of a do- is chaos rare in nature?” especially in This nonlinear partial differential equa- stretching, non—local vorticity Statistics, University of Idaho), “coefficients” in the expression f(x) ). Fa- effects as well as for non—zero nut: p times along the meridians and q light of the fact that ecological models Robert A. Desharnais (Depart- tion possesses a wealth of conserved mous examples include the “discrete lo- thickness of the filament core (a times around its longitudes, and then glu- abound with chaotic dynamics. With the ment of Biology and Microbiol- quantities: the total length is con- ogy, California State University), gistic” equation that uses f(x) = b x (1 - cx) discussion can be found in {3} ing the ends together. After eating the do- latter opinion, one recognizes that the for- stant in time; neither the total squared Shandelle M. Henson (Depart- and the more biologically applicable 2 κ nut, what is left is a (p,q) torus knot. Torus midable difficulties involved in detect- The curvature (s) of a param- ment of Mathematics, Andrews “Ricker” equation with f(x) = b x e-cx . May curvature, , nor the total torsion can etrized curve measures the knots of every (p,q)-type are realized by ing chaos in ecological data have not University), and Aaron A. King was interested in how the sequence of pre- change during the dynamics, and the in- acceleration of the position rector one-soliton solutions of the Vortex Fila- been adequately overcome by the phe- (Department of Environmental τ Science and Policy, University of dictions changes if the number b changes. tegral of the torsion times the curvature r(s), while the torsion (s) mea- ment Equation: a remarkable way in nomenological methods of time series sures how rapidly the curve California at Davis). Supported More precisely, he was interested in the which symmetry manifests itself. Symme- analysis. Researchers have had to rely square will also remain invari- deviates from its osculating plane- by the National Science Founda- long term properties of the sequence–the τ=0 for a planar curve. tries are also responsible for a very simple on such methods until adequate mecha- tion. ant while the curve evolves, and so will “attractor”–which might be as simple as 4 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 13

tion, namely, the identification of an adequate mathemati- cal model for the particular population involved; a model SOLITARY SMOKE RINGS that provides more than good statistical fits to data. The model needs to be biologically-based, that is to say, de- Annalisa Calini, Associate Professor rived from biological mechanisms known to be important Department of Mathematics, College of Charleston for the dynamics of the population under consideration, and it needs to be quantitatively accurate in its predic- he main successes in solving nonlinear differential tions. Unfortunately models that “work” in this sense are equations come when the equations have lots of sym- rare in population dynamics and ecology. The track record metries, often appearing as conserved quantities, like en- of relating mathematical models to data is not good, par- ergy and momentum. One well-known example is the ticularly when it comes to the successful formulation of Kepler problem for planetary motion around the sun, for hypotheses and predictions that are quantitatively test- which conservation of angular momentum allows one to able. Therefore, a major hurdle faced by Cushing and his solve the equations explicitly. colleagues in conducting a study aimed at documenting a route-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). Hotel Work Session Cultures of flour beetles make an The response to the AWS has been tremendous. Our origi- ideal experimental system for nal proposal called for inviting and funding the expenses such a study for several reasons. of 30 graduate students each year. Last year, we stretched First of all, a great deal is known our funds with matching contributions from various de- about their biology (genetics, partments including Princeton, Harvard, and Stanford, physiology, and behavior). This and were able to fully or partially fund about 84 students. is primarily because they are a There were about 140 participants in all, so it’s clear that significant agricultural pest (that FIGURE 1. The upper graph shows the attractor of the many people find the AWS a must-attend event, even if has contaminated stored grain they have to spend their own money. Ricker equation plotted against the corresponding value of b Figure 1. The equation for planetary motion (with c = 0.01). For small values of b, less than approxi- products since ancient Egyptian mately 7.4, there is a single point lying on the graph, indicat- times) and therefore have been In addition to the AWS, the Southwestern Center grant (a has the group of rotations as symmetries. Correspondingly, ing that the attractor is an equilibrium point. For values of b the object of scientific scrutiny for many decades. Further- “Group Infrastructure Grant” from the NSF) has spon- conservation of angular momentum L reduces the model to a between approximately 7.4 and 12.5 there are two points on more, cultures are easy to maintain, easy to census accu- sored a distinguished lecture series, both at the UA and planar system of equations solvable by quadratures the graph, indicating the attractor is a periodic 2-cycle. For rately, and easy to manipulate. Long time series of census other participating institutions (which are the Universi- increasing values of b, the attractor consists of more and data can be gathered in a reasonable length of time since ties of Texas, New Mexico, and Southern California). At more points (with some exceptions). For b = 30 the time When the equation contains partial derivatives, in certain the beetle produces, under normal laboratory conditions, Arizona, we’ve had lecture series by Ken Ribet, Jim lucky situations, there is an infinite number of conserved series of x plotted against t in this case, shown in the lower a new generation in only four weeks. Important for the graph, exhibits the kind of random appearing oscillations to Carlson, Anand Pillay, David Rohrlich, and Alexander quantities, some of them carrying a direct physical inter- project is that they have a complete insect life cycle (lar- which May refers in his seminal paper. Goncharov. The Center also sponsors a post-doc (currently pretation, most coming from more abstract symmetries. vae, pupae, and adult) with inter-stage interactions (can- Shuzo Takahashi), provides summer support and travel Among these equations is the famous Korteweg-de Vries nibalism) that induce nonlinear effects. These nonlinear for graduate students, and has bought computer equip- (KdV) equation, a model of water waves in shallow water, an equilibrium point (the sequence’s limit point, if it effects, which biologically regulate population density ment for the local principal investigators. should happen to converge) or a periodic cycle (a 2-cycle, even when a continual supply of food resources is avail- for example, in which the sequence oscillates alternately able, provide the necessary ingredients for dynamic The local PIs on the grant are Minhyong Kim, Bill between two different numbers.) Or, as May discovered, complexity. McCallum, Dinesh Thakur, and Doug Ulmer. The outside the attractor might be a very complicated oscillation in- PIs are Alex Buium at the University of New Mexico, distinguishable from random noise. Which attractor re- The mathematical model devised by the “Beetle Team” pre- Wayne Raskin at USC, and Felipe Voloch at UT Austin. sults, depends on which value of b is used in the formula dicted population numbers from one census time to the We’ve benefitted from the collaboration of many other f(x). Changes in the type of attractor caused by changes in next, in a manner similar to the famous Ricker equation mathematicians, and the tremendous efforts of Sandy b are called “bifurcations” and the graphical summary of except that each data point consists of three numbers in- Sutton who handles all our administrative work. these bifurcations depicted in Figure 1 for the Ricker equa- stead of one, namely, counts of each life cycle stage. The tion has become a virtual icon of chaos theory. It shows a model uses larval, pupa and adult stages as its state vari- For more about the activities of the Southwestern Center, progressive complexity in attractor dynamics as b is in- ables so as to account for the dominant biological mecha- especially the upcoming Arizona Winter School (March creased, a “route-to-chaos.” nism that drives the nonlinear dynamics of the beetle cul- 9-13, 2002), see our web site at tures — the cannibalism practiced by individuals in one http://swc.math.arizona.edu/ Will a real biological population exhibit a sequence of dy- stage on those of another. The “LPA model” is three di- namic bifurcations, and a route-to-chaos, like those pre- mensional, unlike the one-dimensional Ricker model, and dicted by such simple mathematical equations? There is a therefore is more difficult to analyze mathematically. While Figure 2. Interaction of two solitons of the KdV equation. demanding prerequisite required for answering this ques- some mathematical results have been obtained, there re- Frames are shown for four consecutive values of time 12 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 5

Motivation is as much the responsibility of the student as main many interesting unsolved problems associated with model predicted route-to-chaos. Two years later the team the professor. THE SOUTHWESTERN the model. Nonetheless, the simple recursive nature of the announced the results in the journal Science. In that same three equations in the model makes the model easy to study issue, an editorial remarked that the Beetle Team had pro- In addition, Lesch and I began looking at a problem that CENTER FOR using computers. vided “the most convincing evidence to date of complex Friedlander told me about before my trip. Let us consider dynamics and chaos in a biological population.” and that a domain in R2 which is invariant when acted upon by a ARITHMETICAL In order to connect the LPA model to real beetle data it was “ecologists at last have a convincing example of chaos group. Suppose in addition that the action of this group necessary for the team to “parameterize” or “calibrate” that they can use as a base to understand better complex commutes with the Laplacian, where we are considering ALGEBRAIC GEOMETRY the model. This means numerical estimates for the param- dynamics in other laboratory systems and, more impor- Dirichlet boundary conditions. Then the eigenspaces of eters appearing in the model (of which there are six) needed tantly, in the field.”8 Furthermore, this project demonstrated the Laplacian are representation spaces of this group. One to be obtained and the resulting fitted model shown to be quite clearly that a real biological population can indeed now asks a question involving the multiplicities of the Douglas Ulmer, Associate Professor accurate. This statistical effort required a description of traverse a route-to-chaos as predicted by a “simple” (i.e., irreducible representations occurring in the eigenspaces. Department of Mathematics the inevitable deviations of real data from the model pre- low dimensional) mathematical model, providing a direct Results along these lines have been obtained for domains dictions. In general such deviations, or “noise,” can arise verification of May’s famous tenet. with dihedral symmetries. Much of my time in Köln was hat brings one hundred or so top graduate stu- from a variety of different causes. In the case of data from spent trying to understand these results and attempting dents and a handful of world experts in number the laboratory cultures of flour beetle, however, there is While the demonstration of to understand the problems with using similar techniques theory to Tucson during the second week of March each virtually no census error and, because of the highly con- chaos in a biological population to get results for cyclic groups. year? It could be the beautiful Sonoran desert weather trolled laboratory conditions, there is minimal “environ- was a milestone for the Beetle that time of year, but a more likely reason is the Arizona mental noise,” i.e., few external random disturbances that Team, it was in many ways just I also spent some time in Aachen, a town about 40 miles Winter School! significantly affect the populations as a whole (such as the beginning of its study of non- west of Köln on the border of Germany, the Netherlands, changes in temperature, humidity, light, or food resources). linear population dynamics. The and Belgium where Tom Hoffman, another Arizona gradu- The Arizona Winter School (AWS), which is one of the The source of the noise is “demographic,” that is to say, flour beetle system, with its vali- ate student studying representation theory of finite groups, main activities of the Southwestern Center, is an annual results from random differences among individuals in the dated mathematical model and its was visiting the Nordrhein-Westfalen Institute of Tech- meeting for advanced graduate students and post-docs in populations–differences that are ignored by the model statistical and laboratory compo- nology (RWTH). Hoffman’s advisor and Arizona profes- arithmetical algebraic geometry (or, more briefly, modern (gender, chronological age, body size, egg laying rates, mor- nents, constitute a powerful tool sor, Klaus Lux, was also in Aachen and I spent some time number theory). It has quite a different flavor from a tradi- tality rates, etc.). The team had to devise mathematical with which to delve into the fas- grilling them about representation theory while we hap- tional conference—rather than having a loosely related ways to describe this precise kind of “stochasticity” within cinating intricacies of nonlinear pily drank our fill (and sometimes a wee bit more) of Ger- sequence of talks, with each speaker reporting on his or the beetle system and then incorporate demographic noise theory and to discover new phenomena that are more than man beer. her recent work, the AWS features a tightly integrated set into a stochastic version of the LPA model. With this model just mathematical theories. of courses by leading and emerging experts on a theme of in hand, a large number of statistical techniques became The end of my trip was spent in Edinburgh, Scotland, at a current interest. The program has included such lumi- available for both the parameterization and the validation The “chaotic” beetle populations from the route-to-chaos conference entitled “Progress in Partial Differential Equa- naries as Pierre Deligne, Barry Mazur, Karl Rubin, and of the model. experiment are still being maintained. Data from these tions.” The conference took place in the historic Royal Peter Sarnak, as well as local talent like Bill McCallum populations is now nearly seven years–nearly ninety gen- Society of Edinburgh and I got the chance to listen to many and Doug Ulmer. In the mid 1990’s the team successfully demonstrated the erations–in length. This long time series of data permits good talks by people from all over Europe and the U.S. accuracy of the LPA model in preliminary tests that used the Beetle Team to carry out a detailed study of a biological One of the highlights of the conference for me was a talk A characteristic feature of the AWS is that great effort is existing historical data sets for flour beetle populations. A population exhibiting chaotic dynamics. Chaotic dynam- by Nikolai Nadirashvili from the University of Chicago. expended to make sure that there is significant interaction more important test of the prediction capability of the model ics, while appearing random in their oscillations, none- His talk was entitled “New Results and Open Problems between participants at all levels. For example, each followed when the team designed and successfully imple- theless have discernible temporal patterns. They also paint in Elliptic PDE,” in which he discussed the current state speaker is assigned a group of students who work together mented a yearlong experiment that documented a predicted distinctive geometric patterns in state space. It is rather of elliptic PDE, including the problem I had been working on a project related to that speaker's course, and the stu- period doubling bifurcation. This bifurcation is like that amazing that the Team has been able to document the oc- on in Köln. In addition to the talks, the conference was a dents report on their projects at the end of the School. occurring in the Ricker model except in this preliminary currence of many such patterns predicted by the LPA chance to meet European mathematicians and form con- There are evening working sessions in the hotel (often last- experiment the prediction did not include chaos. None- model–subtle patterns involving not only “attractors” tacts. Saadia Fahki of the Université Paris XII graciously ing until midnight) and several social events. The Winter theless, this work was praised in an editorial appearing in (around which most ecological theory revolves) but un- offered to show my wife and me around Paris if we should schools have led to collaborations between researchers and the journal Nature7 (where the experiment was announced stable entities (called “saddles”). Furthermore, in a study ever make it there. students at various universities and to a real sense of ca- by the Beetle Team) as “an unusual blend of nonlinear to appear in the journal Science later this year and led by maraderie among the next generation of number theorists. dynamics theory, statistics and experimentation” with re- the newest Team member, Professor Shandelle Henson In sum, my trip was an unforgettable experience. I saw sults that “are of uncommon clarity for ecology” in show- (former Hanno Rund Visiting Professor in the Department parts of Europe that I had never seen, and even learned Another unique aspect of the AWS is the “professional ing “how the marriage of nonlinear models and experi- of Mathematics), unusual patterns in the data are shown “ein bisschen Deutsch.” I met other mathematicians with development component.” This part of each AWS is meant ments can help” ecologists accomplish the task of under- to be “lattice effects,” i.e., caused by the fact that animals whom I sincerely hope I will maintain at least a friend- to give students some training in an aspect of the profes- standing and anticipating the consequences of environ- come in whole numbers (mathematical models, like the ship, if not a working relationship. As I look back upon sion beyond mathematics. The professional development mental perturbations. In short, the Beetle Team had pro- LPA model, deal with mean numbers). This is yet another everything that I saw and did, I realize I have only begun components have ranged from a workshop on teaching, vided an example in ecology “of a model that actually lives source of potentially predictable patterns in ecological data to reap the benefits of my time abroad. (including viewing a tape from the Third International up to [its] promise.” whose occurrence was observed for the Mathematics and Science Study -the famous TIMSS study 7 Peter Kareiva, Nature 375, 1995, first time in real data by the Team. comparing math education in various countries- with a With the preliminary work of deriving, 189 panel discussion), to sessions on mathematical writing, analyzing and calibrating a model fin- These, and other studies, support a gen- jobs in industry, and symbolic computation. ished, the Beetle Team was set to design 8 C. Godfray and M. Hassell, eral tenet that has evolved from the and implement an experiment to test a Science 275 (1997), 323 Team’s research. It is unlikely that one 6 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 11 can provide adequate explanations of ecological data by always support a classical tenet of ecology concerning reversed. It seems that foreign mathematicians have vis- means of model attractors alone (or even fuzzy stochastic competitive exclusion–a principle on which the notion of ited for a few weeks or a few months, but hardly any seem versions of attractors). Instead, one must expect to observe ecological niche is based. This is particularly interesting to have visited for a whole year or more. patterns attributable to a variety of deterministic entities, because flour beetles were used in early studies that sup- both stable and unstable and also transient and asymp- ported that now widely accepted theory. However, there I’m sure it will only get easier in the future to arrange such totic, all mixed into randomly occurring episodes by were anomalies in those early experiments that Dr. visits; Chinese mathematicians want more visitors and stochasticity. Nonetheless, what the Team has also shown– Edmunds hopes to explain and with this as a starting often asked for my ideas on how to attract them. My posi- in at least one laboratory system–is that this rather daunt- point the Team hopes to conduct experiments that lead to tion was not advertised and didn’t even exist until I asked ing mix of complexity can be sorted out by means of a low new insights into competition theory. about it, but all it took was the initiative to ask and a good dimensional mathematical model. word from my advisor. A good place to start asking might The interdisciplinary projects carried out by the Beetle be the mathematical division of the Chinese Academy of The Beetle Team’s decade long project on complex dynam- Team have brought mathematical models into a closer con- Sciences in Beijing, since they do have a lot of short-term ics and chaos in population dynamics is the subject of the nection with population biology and ecology. One of the visitors at the moment. Even without any Chinese lan- inaugural book in a new series on Theoretical Ecology, Team’s hopes is that its successes not only provide in- guage, life in a major Chinese city is quite easy to adapt to, scheduled to be published by Academic Press next sum- sights into nonlinear population dynamics, but that they and one could always ask the host institution to organize topology for advanced undergraduates. The latter was mer. The Beetle Team has also carried out other experi- provide a modest step towards the “hardening” of eco- a crash course in basic Chinese. I am certain that any designed to prepare undergraduates for a lecture series to ments designed to delve further into the complexities of logical science–a step towards raising its explanatory future mathematical visitors to China will encounter the be given by Bruno Harris, who will be visiting Nankai chaos. For example, an experimental documentation of a power beyond purely theoretical speculation and a satis- same warm-heartedness, generosity and hospitality in his this fall. Dr. Harris’s visit grew out of a conference in Tianjin dynamic property called “sensitivity to initial condi- faction with only qualitative accuracy, reasonable “guess- or her daily interactions that made our visit so easy and so in memory of K. T. Chen and W. L. Chow; Dr. Harris will be tions”–widely recognized as the signature characteristic timates,” and verbal metaphors. It is true that the labora- memorable. lecturing on Chen’s iterated integrals. I tried to get the of chaotic systems–was recently published by the Team in tory system used in the flour beetle experiments is a rela- students as quickly as possible to a basic understanding the journal Ecology Letters. In another example the Team, tively simple biological system, and that the low dimen- of differential forms and integration on manifolds so that together with Dr. Aaron King (a former student of W.M. sional LPA model is a simple mathematical model of that I could touch on the idea of differential forms and De Rham Schaffer and also a Flinn Postdoctoral Fellow in the Ap- system. Nonetheless, we can find motivation and inspira- cohomology on path spaces. It was a challenge but the A GRADUATE EXPERIENCE plied Mathematics Program at the University of Arizona), tion for the study of simple systems from May’s seminal students were very good, most of them having been ac- has recently completed new experiments designed to study 1976 paper: IN GERMANY cepted to Nankai after winning a national math sublte model predicted, temporal patterns within the cha- competition. otic dynamics of the beetle cultures (patterns involving “Not only in research, but also in the everyday world complicated quasi-periodic motion with various rotation of politics and economics, we would all be better off Jeff Selden, Graduate Student Beyond teaching and research, we really came to enjoy numbers associated with unstable cycles embedded if more people realized that simple nonlinear sys- Department of Mathematics daily life in Tianjin. We lived on campus (as do most fac- within the chaotic attractor). tems do not necessarily possess simple dynamic ulty there); we would begin most days with an early walk properties." spent the month of June working with Matthias Lesch around campus and then get breakfast at a local outdoor The Beetle Team, with its unique interdisciplinary blend at the University of Köln in Germany. Lesch, who stud- market. We did essentially all of our shopping at open-air of mathematics, statistics, and experimentation, has not ies global analysis and geometry, was an Associate Pro- markets, which were much more convenient and acces- restricted its studies in nonlinear dynamics to chaotic dy- fessor in the Department at the University of Arizona until sible than your average grocery store in the U.S. Every- namics. Past studies have included investigations of mul- CENTER FOR January, 2001, when he took a position at the University of thing we ever needed was within an easy (and safe) walk tiple attractors, resonance in periodically fluctuating habi- Köln. I began working with him in the spring of 2000 or bike ride from home. tats, phase shifting, and saddles and their stable mani- RECRUITMENT & when I was preparing for the oral preliminary examina- folds. Characteristic of these studies is that they have, in tion, which entails studying and then presenting a talk on RETENTION OF Food was a highlight of our time in China, and it will some cases, provided explanations previously unavail- a current research paper. After he left, I started working probably be a while till we dare to go to a Chinese restau- able for patterns that had been observed in data and, in with Lennie Friedlander, a Professor in the Department, SECONDARY MATH rant in the U.S. Ultimately, though, it was the social atmo- other cases, the identification and documentation of un- who studies spectral theory and geometry. In order to keep sphere of our many wonderful Chinese meals that made expected and non-intuitive phenomena predicted by the TEACHERS a working relationship with Lesch, I applied for and re- them so special. The best meals were at friends’ homes LPA model. ceived a VIGRE grant which covered the airfare for my trip and most special of all were the meals that we were lucky to Europe. enough to share with Professor Chern, a kind, cultured Currently studies are underway that extend the Team’s Fred Stevenson, Professor and very gracious host. Several times during the year we research in several new directions. For example, a modifi- Department of Mathematics Upon arrival in Germany, I met Lesch’s doctoral student, were invited to dinner at his house and it was always a cation of the model has been formulated to include the Christian Frey. Together we studied K-theory for C*-alge- great honor. When Chern turned 89 the Institute arranged genetics involved in flour beetle adaptation to the insecti- he University of Arizona is establishing a Center for bras and I gave two introductory talks on the subject to a a festive birthday dinner at a fancy campus restaurant cide Malathion. An experiment is under way that tests the Recruitment and Retention of Secondary Mathemat- group of pre-diploma students in Köln. Working with with many friends and mathematicians from around prediction of this model that the beetle dynamics will un- ics Teachers in the Department of Mathematics. The Cen- Christian in preparing my talks was an invaluable experi- Tianjin and Beijing; the dishes were too numerous to keep dergo a dynamic bifurcation as the population genetically ter is in response to a critical shortage of qualified second- ence; not only for the mathematical benefits of collabora- track of but I do remember delicious crabs. adapts to the insecticide. Another example involves a mul- ary mathematics teachers. This crisis is national in scope; tion but for the education I received in the German student Perhaps the most surprising thing about our visit was that tiple species version of the LPA model studied by Dr. Jeff experienced teachers are retiring or quitting in droves; it attitude. German students do not have to pay to go to it was so novel. Chinese mathematicians regularly experi- Edmunds, recent graduate of the Department of Mathemat- is estimated that 35% of new mathematics teachers leave university. While they are there, no one forces them to do ence the mathematical environment in the U.S. and bring ics. In his thesis, Dr. Edmunds showed that two compet- the profession within five years and 30% of University homework, attend class or study. It is up to the individual back new perspectives to China, yet the situation is rarely ing flour beetle species, placed in the same habitat, do not students who get a teaching degree in mathematics do not student to decide why they want to study a certain subject. 10 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 7 lating intellectually and exciting culturally. My wife and I enter the classroom at all, but opt instead for the higher We also plan a large scholarship initiative to attract young made close friends, enjoyed the small pleasures of daily salaries and prestige of other professions. We mirror the students into teaching. Already this fall we have secured life in a Chinese city, ate incredible food, and accumu- situation here in Tucson. Ten years ago the University of funding to provide 13 scholarships to juniors and seniors lated quite a lot of frequent train rider miles travelling Arizona would provide the school system with at least 15 at the University who are considering a teaching career. around China. new mathematics teachers per year; this year we have only In the future we hope to have as many as 25 scholarships two on track to graduate. and include high school seniors and university freshmen During the last year of my doctoral program, while send- and sophomores as recipients. ing off a huge stack of job applications to universities all The shortage of qualified mathematics teachers has been across North America, I wondered what options were out exacerbated by a recent mandate from the State Board of In the near future we intend to initiate a program where there to do mathematics elsewhere. I asked Alan Weinstein Education. Last year the Board ordered that all entering new teachers and experienced teachers can link together, at Berkeley for his thoughts on this. First he asked how my high school students must take two consecutive years of get to know each other, and provide mutual support, en- French was, and when I replied that it was pretty rusty mathematics covering material that has traditionally been couragement, and advice. We hope that this will alleviate but that I had studied some Chinese in college, he imme- taken only by the better math students. In most schools the exodus from the classroom by first and second year diately said something to the effect of, “Well, just ask this meant that all students were to take Algebra 1 their teachers. We also want to provide a professional compo- Chern, he can probably send you to China.” freshmen year and Geometry their sophomore year. Unfor- nent to the life of the classroom teacher. This involves other countries; the theme was usually differential geom- tunately middle schools cannot prepare all students for funding teachers who wish to return to the classroom and Shiing-Shen Chern is a Chinese mathematician, an emeri- etry and Chern was always an active participant unless Algebra 1 in the 9th grade. The failure rate for freshmen take advantage of our graduate courses in Mathematics tus professor at Berkeley and one of the great modern dif- he was out of town. was expected to be bad and it turned out to be embarrass- Education and also provide support for those who wish ferential geometers, so with some trepidation I sent him a ing: in many high schools more than half the students to attend and participate in regional and national profes- letter asking how one might arrange a mathematical visit I often visited Peking University, an hour and a half train failed freshman mathematics. This year, those failed stu- sional meetings. to China. First he called my advisor for a background ride away, to give or listen to talks, and I attended several dents are in 10th grade Geometry, and failure rates may check, and then he called me and said, “So, you want to international conferences around China. The most memo- very well be even higher. The mandate was made neces- This is an expensive undertaking. Setting up the Center go to China? You should go to my institute.” Chern rable was a symplectic geometry conference that spanned sary because of the creation of the AIMS test (Arizona In- requires finding space, hiring directors and staff, and pro- founded the Nankai Institute of Mathematics in 1985, af- three cities: we started in Chengdu, the capital of Sichuan strument to Measure Standards), the Arizona state high viding for operating funds. Carrying out our plans is ter China opened up to the west and Chern began visiting province, then moved to Tianjin and finally ended up in school exit examination. This test is first given to all stu- costly; in fact, elementary mathematics will show that the China again after so many years away. Now, at the age of Beijing. I gave a talk on my own work and learned a lot dents at the end of their sophomore year and, on the ad- tutoring initiative alone has a surprisingly high price tag. 89, he lives nearly full time in Tianjin, on the Nankai Uni- from an excellent mix of mathematicians from all across vice of lawyers, it was decided that all mathematics tested We have begun operations thanks to generous gifts from versity campus, and he is tremendously respected through- China together with prominent symplectic geometers and by AIMS must be presented to the students before they take private individuals. These funds, along with some state out China. It is not uncommon to meet taxi drivers in China topologists from Japan, the U.S. and Europe. the test. The test includes Algebra 1 and Geometry. Natu- funds, have given the Center the chance to carry out its who know his name. Therefore, it is not surprising that rally schools are scrambling to salvage students by offer- plans for the first year and even begin outlining plans for once Chern took to the idea of my visit to Nankai, the rest The mathematical content was stimulating enough but just ing supplementary remedial courses but finding qualified the second year. We have hired two of Tucson’s finest of the arrangements went very smoothly. in case we needed more stimulation, we were served three teachers is a virtual impossibility nowadays. teachers to direct operations of the Center. Sue Adams banquet-style meals each day; in Chengdu the food was has been a mathematics educator for over 30 years in the When I was offered a three year postdoctoral position at the best, Sichuan being particularly well known for its The Center is attempting to help with this difficult reality. Tucson area. During the past two decades she has served the University of Arizona, the hiring committee was kind cuisine. At our final lunch in Chengdu, a meal of Sichuan- Our first job is to address this emergency in the secondary as Head of the Mathematics Department at University High enough to let me take one year off to spend in China. Fur- style dimsum, the tables had motorized lazy susans that mathematics classrooms. This fall we are piloting a pro- School and then as High School Mathematics Coordina- thermore, my wife was kind enough to agree to come with seemed to magically sprout new dishes every time you gram employing 20 tutors from the University and Pima tor for Tucson Unified School District. Ann Modica has me, and so all things conspired to make the trip a reality. looked away. It was a tough meal to keep up with, and College and 15 cooperating teachers from two middle also been a mathematics educator in Tucson for over 30 After a first year and a summer in Tucson, we set off for when we got back to Tianjin we skipped several meals just schools and six high schools. The tutors are taking a one years. She has won a number of prestigious teaching China in September 2000. to rest our stomachs. credit hour course and receiving $10 per hour for their awards including the Presidential Award for Excellence tutoring. This is only a beginning; bear in mind that there in Science and Mathematics Teaching. For the past de- My work is in low-dimensional symplectic and contact I taught two classes each semester, all quite a change from are close to 60 secondary schools in Tucson and several of cade she served as Head of the Mathematics Department topology; while no one at Nankai works in precisely this my teaching routine in Tucson. The first semester I taught them have at least 100 students who need and want extra at Rincon High School. And thanks go to Professor Steve field, there were many researchers and graduate students basic graduate differential topology (with six students) help in mathematics. In the spring semester we begin a Willoughby of the Mathematics Department who has pro- eager to learn what they could about the subject and to and a special “Mathematical English” course (with 30 stu- second program placing as many as 40 high school stu- vided a temporary home for the Center in his office. This teach me about their own related work in geometry, topol- dents). Differential topology was lots of fun and we be- dents in middle schools and high schools. In the second is an especially generous offer in our space-starved Math- ogy and analysis. came good friends with all of the students. The English year we hope to expand our programs even further, per- ematics Building. course was more difficult; I didn’t really feel qualified but haps involving as many as 80 tutors and 40 coordinating The Institute has a well-endowed mathematics library, so obviously none of the real English teachers on campus teachers. This is not only an initiative to help students The Center has the support of the University Administra- that access to books and journals was never a problem. were qualified either. I decided to focus on oral skills, since and teachers in the schools but it is also a recruiting tool. tion. While it is not yet incorporated into the University Early on in our visit a colleague, Fuquan Fang, and I orga- the students seemed pretty good at reading and writing. There are many university students who have an aptitude structure we have been given assurances that if the Center nized a working seminar on the topology of contact struc- We ended up swapping short lectures on elementary top- for mathematics and who love to help others. We want to can make a tangible difference in the next few years the tures and foliations with several graduate students and a ics, so that they could all practice listening to lectures in give them the opportunity to experience what it is like to University will provide necessary resources and fund it few other researchers. Almost all mathematical discus- English as well as giving presentations. use their talents in a meaningful way. long term. We are hopeful that the work of the Center will sions happened in English, since my Chinese is not good make a clear difference in the near future so that we can enough for mathematics. The Institute had weekly collo- The second semester I taught “Topics in low-dimensional continue this work. quia with speakers from all over China and often from geometric topology” and a special course on differential 8 Mathematics Spring 2002 The University of Arizona The University of Arizona Mathematics Spring 2002 9

croscope, take a tour of IBM and spend time selecting sci- ENRICHING HIGH A MYOPIC VIEW OF ence books at Borders bookstore. SCHOOL PROGRAM ACCESS The schedule for the Camp ACCESS 2001 can be seen at http://www.math.arizona.edu/~dsl/camp2001.htm. MATHEMATICS OR HOW SOME STUDENTS Photos are visible at http://w3.arizona.edu/~access/. CHANGED MY LIFE I have been involved with a number of rewarding and Elias Toubassi, Professor, worthwhile activities in my 40-year career as a faculty mem- Department of Mathematics David Lovelock, Professor ber at universities around the world, but this has been the Associate Head for the Entry Level Program Department of Mathematics most rewarding and the most worthwhile. I can do no better than quote from a letter from one of the camper’s he Mathematics Department, in cooperation with n Fall 1990 I was teaching honors Calculus 1 in our parents, the full text of which can be read at http:// local school districts, has developed a two-year mathematics computer classroom. We had just opened w3.arizona.edu/~access/letters/letters.html. (You may project to work with high school mathematics teachers to this room – the first in the U.S. – and had made it acces- Grocery store botany need a Kleenex while reading the full letter. I do.) develop curriculum material that helps mathematics teach- sible to students in wheelchairs, even though in my previ- do. They turned to Georgia Ehlers, the Coordinator of ers assist their students to meet the State standards as ous 30 years of teaching, I had never had a student with a Grant and Scholarship Development in the UA Graduate “I have thought a lot about what it is that made measured on the AIMS test. This collaboration has re- physical disability in any of my classes. Imagine my sur- College. She advised them to find a faculty member to your camp so wonderful for my son Justin as well sulted in an Eisenhower grant proposal which was prise when one of the first students to enter the room was spearhead the project. as his fellow campers. To say you changed my son’s funded by the Arizona Board of Regents. The first work- in a wheelchair. This was John Olsen, a freshman in the life is not hyperbole. When I brought Justin to camp shop was held in June 2000 and the second in June 2001. Department of Electrical and Computer Engineering. Some To cut a very long story short – and despite all my kicking 2 weeks ago, I entered with a disabled child. Two The participating districts are: Amphitheater, Catalina will remember him tearing around campus with a Mickey and screaming explaining that I knew nothing about dis- weeks later, I am going home with an able child Foothills, Flowing Wells, Marana, Sunnyside, and Tuc- Mouse flag attached to the top of an antenna fixed to his ability issues – I became that person, and so Program AC- who happens to have challenges in life. He has gone son Unified School District. electric wheelchair. CESS (Accessing Career Choices in Engineering and the from withdrawn and frustrated to friendly with ScienceS) was born, and my life was changed forever. other children and excited about learning. You have taught him more in 2 weeks than you will ever The 2001 workshop was held from June 4 through June 15 John and I remained in touch after he finished Calculus 1. know.” on the campus of the University of Arizona and: John graduated and started postgraduate studies in ECE. With the help of an interdisciplinary cross-campus group, Then one day in 1997 he introduced me to another gradu- Ali, Georgia, John, and I wrote the proposal, and the NSF By being frugal, and accepting donations, we will be able •Provided a forum for teachers to create curriculum ate student, Ali Mehrabian, in Civil Engineering. Ali also funded the $500,000 three-year request in July 1998. It is a to run Program ACCESS for an extra year beyond the end material that addresses the State standards in math- whips around campus in an electric wheelchair. The two multi-faceted project aimed at students with physical dis- of the initial three, until summer 2002. After that, the fu- ematics; of them then told me a story. abilities from middle school to graduate school. ture is unclear, but of all things we have done, we will try •Included instruction on mathematics topics such as Ali and John had been remorsing over how few students to keep the camp going. Finally, I should report that John probability, statistics, and logic, that give teachers a We have undergrads mentoring school kids, and gradu- with physical disabilities were studying Science, Engi- now has his Ph.D. and is working for IBM. Ali is close deeper background in standards-related topics; and, ates mentoring undergrads –all with physical disabilities. neering, Mathematics, or Technology (now known as behind. •Focused some sessions to develop new ideas to teach We give encouragement grants to school teachers to help SMET) at UA. They had done a web search, and stumbled traditional subjects such as algebra and geometry. them make their classes more accessible. Ali and John Thank you John and Ali for twisting my arm. upon a National Science Foundation (NSF) request for pro- give in-class demonstrations at schools to show that their posals to encourage more students to study SMET. They In addition, participating teachers took part in two full handicaps are not handicaps. We work with the College wanted to write such a proposal, but didn’t know what to days of in-services during the 2001-2002 academic year. of Architecture and have their sophomores analyze vari- The Fall in-service was held at Sahuaro High School in ous labs on campus and suggest ways to make them more A YEAR IN CHINA October and the Spring in-service will be held at Flowing accessible. And we offer a summer camp (Camp ACCESS) Wells High School in March, 2002. The highlight of the for 12 middle school kids. David T. Gay, Visiting Instructor, Postdoctoral Scholar in-services is time when teachers share how the new cur- Department of Mathematics riculum lessons are being received by students in their The camp is our pride and joy. It lasts for 2 ½ weeks, and is classrooms. These programs result in the formation of run by Faith Bridges, an adjunct faculty member in the just returned from a year of teaching and research at community support among faculty and high school mathematics department, who has experience as a middle/ Nankai University and the Nankai Institute of Math- teachers. high school teacher. The camp emphasis is on science, not on disabilities. The program is a smorgasbord of hands- ematics in Tianjin, China; this article is a report on my mathematical and cultural experiences there. For a good Participating teachers received a $400 stipend, a small on activities and is presented by UA faculty from many number of mathematicians in the U.S. (those originally curriculum allowance, and a classroom set of graphing disciplines. These activities range from Grocery Store from China) this story won’t have too many surprises but calculators. The following faculty assisted with the work- Botany to Mammals With Wings, from Being a Paleontolo- it might be entertaining to hear it told from the point of shop: Christopher Goff, Brigitte Lahme, Carl Lienert, Bin gist to How Spaceflight Affects Humans, from Practical view of someone not Chinese. To those who have never Lu, Jerry Morris, Diann Porter, Maria Robinson, Joseph Dendrochronology to Experiments With Liquid Nitrogen, been to China or even thought of visiting, I hope the story Watkins, and Stan Yoshinobu. from Hands-On Math to Blood and Guts!, from Building and Testing Model Bridges to What’s Inside My Computer? is interesting and encourages more foreign mathemati- The campers visit a clean room, play with an electron mi- cians to visit China in an academic capacity. Building bridges - with toothpicks and jellybeans It was a wonderful year: productive mathematically, stimu-