The following awards were presented at a special ceremony held on Sunday, August 1 at the Providence Mathfest:
Carl B. Allendoerfer Awards
Trevor Evans Awards
Lester R. Ford Awards
Merten M. Hasse Prize
George Pólya Awards
Carl B. Allendoerfer Awards
The Carl B. Allendoerfer Awards, established in 1976, are made to authors of expository articles published in Mathematics Magazine. The Awards are named for Carl B. Allendoerfer, a distinguished mathematician at the University of Washington and President of the Mathematical Association of America, 1959-60.
Victor Klee and John R. Reay
Surprising but Easily Proved Geometric
Mathematics Magazine, Vol. 71, No. 1, February 1998
The problem addressed in "Surprising but Easily Proved Geometric Decomposition Theorem" is illustrated whimsically by the drawing on the cover of the issue of Mathematics Magazine in which it was published.
Two planar sets A and B are homothetic if they are similar and similarly oriented. Two sets A and B are 2-homothetic if each can be partitioned into two disjoint sets (A = A1 union A2 and B = B1 union B2 with A1 and A2 disjoint and B1 and B2 also disjoint) in such a way that A1 and B1 are homothetic and A2 and B2 are homothetic. The authors prove the surprising result: Two sets in the plane are 2-homothetic provided each of them is bounded and has nonempty interior. The proof follows easily from a strengthened form of the Cantor-Bernstein Theorem. Though accessible to undergraduates, the article ends with a set of open problems, and draws upon a wide variety of important mathematical work by such mathematicians as Banach, Bernstein, Cantor, Fraenkel, and Tarski. Its fun, surprising, crystal clear, and somehow both straightforward and non-trivial at the same time.
Victor Klees 1949 Ph.D. was from the University of Virginia, which attracted him because of an initial interest in point-set topology. While there, he became interested also in functional analysis and convex geometry. After the move to Seattle in 1953, his interests broadened to include combinatorics, optimization, and computational complexity. These days, he says he likes to work in a variety of fields in order to spread his mistakes more thinly. He is a co-author, with Stan Wagon, of the MAA book Old and New Unsolved Problems in Plane Geometry and Number Theory. Professor Klee was MAA President in 1971-73.
John R. Reay studied music at Pacific Lutheran University and mathematics at the University of Washington, where Victor Klee directed his 1963 Ph.D. thesis. He now teaches at Western Washington University, and plays in the Whatcom Symphony Orchestra. The joint paper on 2-homothetic sets grew out of a talk he prepared for the Visiting Lecturer Program of the MAA, and the goading of friends who wanted a written version. The talk was based on earlier lectures of Klee.
Response from Victor Klee
The recognition is especially pleasant, since Carl Allendoerfer was chairman of the University of Washington Mathematics Department when I joined the department in 1953, and was still chairman when John Reay received his Ph.D. there. It was fun working with John on the paper, and were both indebted to Jack Robertson for the initial suggestion that led to the paper.
Response from John R. Reay
What higher honor can a mathematics teacher be given, than to have his or her presentation be described as ... crystal clear, and somehow both straightforward and non-trivial at the same time? That is what all teachers strive for.
Donald G. Saari and Fabrice Valognes
Geometry, Voting, and Paradoxes
Mathematics Magazine, Vol. 71, No. 4, October 1998
Combinatorics is the traditional technique to compare and analyze voting procedures. What makes this paper so inviting is the use of geometry to compare voting procedures. Accessible, easily visualized mathematics leads to some surprising results. After seizing the readers interest with simple examples, the authors present some historic voting procedures (the Borda Count and the Condorcet winner). Then, in the authors words, they demonstrate how geometry dramatically reduces these previously complicated issues into forms simple enough to be presented to students who can graph elementary algebraic equations. The article reveals a novel view of a classic problem.
Don Saari received his Ph.D. from Purdue University and his B.S. from Michigan Technological University in Houghton, MI. He currently is the Pancoe Professor of Mathematics and a Professor of Economics at Northwestern University. His research interests center around dynamical systems and their applications -- primarily as applied to the Newtonian N-body problem and to issues coming from the social sciences. His most recent book is Basic Geometry of Voting, Springer-Verlag, 1995. Because of Saaris current interest in voting procedures, he makes frequent research trips to the French Institute of Social Choice and Welfare, Universite de Caen, where he met his co-author Fabrice Valognes.
Fabrice Valognes was born in 1969 in the city of Caen (Normandy) and received all of his degrees from the University of Caen. In 1998, he received his Ph.D. in field mathematical economics from the same university. Valognes supervisors were Professors Maurice Salles and Dominique Lepelley (both of the University of Caen). However, Donald G. Saari (Northwestern University) and William V. Gehrlein (University of Delaware) also influenced him greatly and quickly became his part-time supervisors. Valognes dissertation is entitled Essays in Social Choice Theory. He is an Associate Professor of Economics at the University of Namur (Belgium).
Response from Don Saari
An important aspect of being teachers and researchers of mathematics is communicating what we have learned -- namely, exposition. Because of the importance I place on exposition, I am delighted and honored to receive the Allendoerfer Award! But, exposition is not a solitary pursuit; it is a form of communication where presentation can be sharpened only through considerable interaction with others, such as the give-and-take between my co-author Fabrice Valognes and me. Authors are particularly fortunate if they encounter a gifted, concerned editor who helps improve the final product. We did, and we are so delighted to publicly thank Paul Zorn, the editor of Mathematics Magazine. We also thank the Selection Committee and the MAA for this recognition.
Response from Fabrice Valognes
I am very proud to be honored by the Mathematical Association of America. You cannot imagine how much of a pleasure it is to receive an award from mathematicians for my work in social choice theory. To put it in a nutshell: What fantastic news!
Trevor Evans Awards
The Trevor Evans Awards, established by the Board of Governors in 1992 and first awarded in 1996, are made to authors of expository articles accessible to undergraduates that are published in Math Horizons. The Awards are named for Trevor Evans, a distinguished mathematician, teacher, and writer at Emory University.
The Youngest Tenured Professor in Harvard
Math Horizons, September 1998, pp. 8-12
In this article Ravi Vakil describes the awesome talents of Noam Elkies, who, in 1993, at the improbable age of 26, became the youngest person ever tenured at Harvard. It is an amazing tale -- from Elkies early love affair with Euclid's Elements (reminding us once again why classics are so named), to his triumphs in the Mathematical Olympiads, to his proofs and counterexamples in number theory. While the focus rests on Elkies achievements in such arenas as chess, music, and puzzling, the article deftly introduces some serious mathematics along the way, the ABC Conjecture being a notable example. Vakil thereby simultaneously educates and fascinates us with this glimpse of one whose mind burns with the brilliance of the noonday sun.
Ravi Vakil was born in Toronto, Canada in 1970. He received his B.Sc. at the University of Toronto in 1992, where he was a four-time Putnam Fellow. He received his Ph.D. in 1997 from Harvard University (in algebraic geometry, under the supervision of Joe Harris). He was then an Instructor at Princeton University, and is now a C.L.E. Moore Instructor at M.I.T. He has long been interested in communicating the excitement of mathematics to others. While in high school, he co-founded the journal Mathematical Mayhem (now a publication of the Canadian Mathematical Society, along with Crux Mathematicorum). He is also the author of the book A Mathematical Mosaic: Patterns and Problem-Solving, part of which was the basis of the article on Noam Elkies.
Response from Ravi Vakil
I am thrilled and honored to be awarded the Trevor Evans Award. I would like to express my gratitude to the Trevor Evans Award Committee.
An often-neglected responsibility of mathematicians is to get across the excitement and substance of mathematics to others. On the whole, we do quite a good job of speaking with younger people within the mathematical community, for example, through excellent magazines such as Math Horizons.
We are less successful with society at large. For some reason, we tend to try to justify mathematics purely with appeals to utility, advertising ourselves as a less useful version of engineering or computer science or medicine. Many important results are misleadingly described in the press as being important because of some tenuous link to some form of technology. For some reason, we dont try to get across the most fundamental message of mathematics: that in trying to understand the universe by asking natural questions, we continually come across fascinating and beautiful structures. In return, by understanding patterns among patterns, we find much that is of use.
We should keep in mind that some of the most popular scientific articles in newspapers are about astronomy, certainly a field without immediate applications. What we do is full of beauty and excitement, and we should be able to convince any interested listener of this fact.
Once again, this award is a true honor for me, and I would like to express my thanks to the MAA.
Lester R. Ford Awards
The Lester R. Ford Awards, established in 1964, are made to authors of expository articles published in The American Mathematical Monthly. The Awards are named for Lester R. Ford, Sr., a distinguished mathematician, editor of The American Mathematical Monthly, 1942-46, and President of the Mathematical Association of America, 1947-48.
Applications of the Universal Surjectivity of
the Cantor Set
The American Mathematical Monthly, November 1998, pp. 832-839
A classical theorem due to Alexandroff and Hausdorff states that every compact metric space is the continuous image of the Cantor set. In this paper Yoav Benyamini presents striking applications of this result to diverse areas of mathematics. Each of these applications involves an existence theorem that Benyamini shows us how to prove using the universal surjectivity of the Cantor set. Some of these results are well known, such as the existence of space-filling curves and the isometric identification of every separable Banach space with a subspace of C([0,1]). Other results are more unusual, such as the existence of a compact convex subset of Rn+2 whose faces include congruent copies of all compact convex subsets of the unit cube in Rn. Other results are even more counter-intuitive, such as the existence of a continuous real-valued function f on R with the property that for every bounded sequence (an) of real numbers, there exists t c R with f(t+n) = an for all n. Benyamini ties all these results together in a pretty package with the common theme that the Cantor set and its universal surjectivity lurk behind many strange phenomena.
Yoav Benyamini is a professor of mathematics at the Technion-- Israel Institute of Technology, in Haifa, Israel. Born in Jerusalem in 1943, he completed his mathematical education at the Hebrew University in Jerusalem, receiving his B.A. in 1966, M.Sc. in 1970, and Ph.D. in 1974. After two years as a Gibbs instructor at Yale and a year at Ohio State University in Columbus, Ohio, he joined the Technion in 1977. Benyamini later visited the University of Texas at Austin for two years, and the Weizmann Institute for one semester. He was chairman of the mathematics department at the Technion in 1993-95, and is currently the vice-provost for undergraduate studies.
Response from Yoav Benyamini
I am honored to be one of the recipients of the 1997 Lester R. Ford Awards for 1999. I looked at the list of past winning articles, and found many articles that I have read with great pleasure during the various stages of my mathematical career. I hope that my contribution, and those of my co-winners, will be a worthy continuation to this impressive list.
Jerry L. Kazdan
Solving Equations, an Elegant Legacy
The American Mathematical Monthly, January 1998, pp. 1-21
The paper discusses various types of equations: polynomial equations in one and several variables, linear and nonlinear differential equations, diophantine equations, and congruences. The overriding idea is that familiar procedures for solving equations, often viewed as tricks, can be seen as belonging to broad themes which, in turn, yield new insights on equations. Among the themes are: exploiting symmetry, finding a related problem, understanding the family of all solutions, finding obstructions when an equation has no solution, using variational methods, and reformulating a problem. An extensive discussion of symmetry is an important unifying thread. It bears on complex conjugation, linear differential equations, Markov chains, Lies Galois theory of differential equations, and Pells equation. Kazdans article is an instructive and wide-ranging tour of the mathematicians workshop in important classes of equations.
After graduating from Rensselaer Polytechnic Institute in 1959, the heart of Jerry Kazdans mathematical education was at the Courant Institute of New York University. For Kazdan, that was an amazingly rewarding experience. After receiving his Ph.D. in 1963, he was a Benjamin Peirce Instructor at Harvard University before moving to the University of Pennsylvania in 1966. Kazdan has been at Penn ever since, punctuated by enlightening visiting positions elsewhere. Kazdans main research interests are partial differential equations and differential geometry.
Response from Jerry Kazdan
I hope this article is useful in giving students a broader, less compartmentalized view of mathematics than is usually presented in standard undergraduate courses. It is a pleasure to thank my friends Dana and Norbert Schlomiuk. They suggested that I give a lecture based on this at the University of Montreal, and then motivated me to write it up.
Note that the article Solving Equations is an excerpt from a longer article that I think is more useful. Although it was too long for the Monthly to publish, one can obtain it from the web at: http://www.math.upenn.edu/~kazdan/solving.html
Polynomial Equations and Convex Polytopes
The American Mathematical Monthly, December 1998, pp. 907-922
How many complex zeros can d polynomials in d variables have? In the case of the bivariate system
a1 + a2x + a3xy + a4 y = b1 + b2 x2y + b3xy2 = 0
with nonzero real coefficients, Bezout's theorem gives an upper bound of six solutions. But it has exactly four. To achieve this better estimate, we use an idea of Newton. To a bivariate polynomial S axuyv, associate its Newton polygon, the convex hull of the vertices (u, v). The mixed area M(P, Q) of two planar polygons P, Q is defined by
M(P, Q) = area(P + Q) - area(P)- area(Q).
In 1975, David Bernstein proved a general theorem that for two equations in two unknowns shows that the number of solutions of a system of two bivariate polynomial equations is equal to the mixed area of the two corresponding Newton polygons. Sturmfels outlines an algorithmic proof devised by B. Huber and himself in 1995 that leads to a numerical approximation for the solution. The author deftly avoids getting overwhelmed by algebraic and geometric detail by using examples and organizing his account of the proof of Bernstein's theorem around three key steps.
The case of real zeros has been seriously investigated for only the past twenty years and little is known. Sturmfels brings us into the cut and thrust of current research with its distance between conjecture and reality and its open questions, leaving much to do that is of interest to combinatorialists, algebraic geometers, and applied mathematicians.
Bernd Sturmfels received his Ph.D. in 1987 at the University of Washington, Seattle, under the supervision of Victor Klee. After postdoctoral years in Minneapolis and Linz, Austria, he taught at Cornell University for six years, before moving permanently to University of California, Berkeley. Sturmfels has been a Sloan Fellow, an NSF National Young Investigator, and a David and Lucile Packard Fellow; he has held visiting positions at New York University and RIMS Kyoto, Japan. He has authored six books and 100 research articles in combinatorics, computational algebra, and algebraic geometry. His latest book on Gröbner basis methods for systems of hypergeometric differential equations (jointly with Mutsumi Saito and Nobuki Takayama) will appear in the fall of 1999.
Response from Bernd Sturmfels
The interplay between solving polynomial equations and convex polytopes is a beautiful subject of mathematics, which can be appreciated by undergraduate freshmen and experts of algebraic geometry alike. I was happy to share some of my excitement for this topic with the readers of the Monthly, and I am deeply honored by this unexpected recognition.
Merten M. Hasse Prize
The Merten M. Hasse Prize was established in 1986 to encourage younger mathematicians to take up the challenge of exposition and communication. In alternate years, it recognizes a noteworthy expository paper that appears in an Association publication, where at least one of the authors is younger than 40 years old at the time of the acceptance of the paper.
The Mercedes Knot Problem
American Mathematical Monthly, Vol. 103, No. 6, November 1996, p. 756-770
The charming article The Mercedes Knot Problem, by Aleksandar Jurisic, begins by asking why extension cords always get tangled. This leads to a puzzle - about twists in elastic cords attached to three walls of a room and a sphere in the middle of the room - which is investigated from many angles as the article unfolds. This delightful diversion turns out to lead to deep and interesting mathematics. Along the way, we learn of connections to knot theory, Pauli's use of spinor calculus to model electrons, a Piet Hein game, braid theory, and machines for twisting electrical wires. The article quotes, as a goal, Hilberts advice that The art of doing mathematics consists in finding that special case which contains all the germs of generality and succeeds wonderfully in realizing that goal.
"The Mercedes Knot Problem" uses a cute visual problem to entice the reader into serious mathematics. It will repay careful study, both for students interested in this corner of knot theory and its numerous connections, and also for students of truly effective mathematical exposition.
Aleksandar Jurisic received a B.A. from the University of Ljubljana in 1987 (working under Jovze Vrabec on applications of topology in combinatorics), and a M.Sc. and Ph.D. from the University of Waterloo in 1990 and 1995 (working under Chris Godsil in the field of algebraic combinatorics). He held a two-year industrial post-doctoral position at Certium Corp., Canada, and the Department of Combinatorics and Optimization at the University of Waterloo, Canada, working in cryptography and algorithmic number theory, and a one-year research position at the Institute of Mathematics, Physics and Mechanics (IMFM) in Ljubljana, Slovenia. Since October 1998, he has been an associate professor at the Nova Gorica Polytechnic and a researcher at IMFM. His main research interests are discrete mathematics and geometry. He loves problem-solving and trying to make difficult things look easy. In his free time he enjoys playing basketball and teaching recreational mathematics.
Response from Aleksandar Jurisic
I rushed into a class to give a lecture, equipped with several colorful accessories (beach balls, long shoelaces, belts, cords, etc.). The unfamiliar crowd of students there told me I must have been in the wrong place. When I persisted that I was on the right floor, corridor, etc., they told me they were certain because it seemed I was preparing something interesting -- and they had never had so much fun in the mathematics class they were waiting for. Actually, it was the Math Club audience across the corridor that was expecting my lecture on knot theory. It was then I decided to show to broader audiences that mathematics and real life are much more connected than we are ready to admit (for example, a periscope and effects of enzymes on circular DNA are related through quaternions).
Thank you for the support such articles are getting through the Merten M. Hasse Prize.
George Pólya Awards
The George Pólya Awards, established in 1976, are made to authors of expository articles published in the College Mathematics Journal. The Awards are named for George Pólya, a distinguished mathematician, well-known author, and professor at Stanford University.
David Bleecker and Lawrence J. Wallen
The Worlds Biggest Taco
College Mathematics Journal, January 1998, pp. 2-17
Dont you hate it when your taco shell shatters and the filling spills out? Things could be worse if the shell were shaped to hold (then lose) as much as mathematically possible. David Bleecker and Larry Wallen take the reader on a delightful trip through an offbeat and appealing optimization problem, with a dash of special functions and computer algebra to add spice.
David Bleecker received his Ph.D. at the University of California, Berkeley with the gracious guidance of S.S. Chern, and with the help of inspirational courses in differential geometry beautifully taught by Blaine Lawson. As an undergraduate, he served on the Stanford tennis team under Dick Gould, who was then the new coach, while Robert Osserman nurtured his interest in geometry. After leaving Berkeley, he joined the faculty at the University of Hawaii where a tremendous inertia has held him there for 25 years. He did manage to write 1.5 books and a number of articles on the application of geometry to physics (e.g. gauge theory and relativity), and he co- authored an undergraduate partial differential equations (PDE) text with his current tennis nemesis, George Csordes. Basic PDEs was recently published by International Press under the auspices of editor and Fields Medalist S.T. Yau, who sat in the very same classes taught by Chern and Lawson at Berkeley.
Lawrence J. Wallen received his undergraduate education at Lehigh University where Everett Pitcher first showed him that mathematics can be beautiful. His graduate work was done at M.I.T. where inspiration came in large part from the beautiful thinking of Witold Hurewicz. Finally, he was seduced by operator theory in the person of Paul Halmos who remains a model and a critic even while their interests diverged. Wallens early work was in operator theory, especially in the structure of operator semi-groups. A little later, he strayed into classical analysis (moment-problems, asymptotic analysis), and finally landed in convexity theory where he remains. Wallen has taught (or tried to anyway) at the University of Hawaii since 1968.
Response from David Bleecker
I am pleased and honored to accept this Pólya Award. I thank my co- author Larry Wallen for having provided the initial idea of the problem and for challenging me to find rigorous existence and uniqueness proofs, as well as numerical approximations. We briefly entertained the notion of studying the higher dimensional problem, but we felt that the casual connoisseur might find hypertacos difficult to digest. Finally, I mention that Joel Weiner noticed that the original problem can be posed in terms of optimizing the energy of a curve with respect to a Lorentzian metric, as in relativity. Relativistic tacos? Now thats fast food!
Response from Lawrence J. Wallen
I am both surprised and delighted that what began life as a problem for a calculus class grew into a little piece that apparently interested several of my colleagues. One citation classified it as amusing -- thats almost as good as interesting. My initial surprise was that Bessel functions arose from such a lowly source and then that other classical functions (elliptic and hypergeometric) appeared in the same way. The optimization problem that we addressed was enjoyably delicate and unstable -- there are lots of big tacos around that look pretty different. Finally, it was delightful to find that interesting applications continue to appear in an eighteenth-century idiom.
Aaron Klebanoff and John Rickert
Studying the Cantor Dust at the Edge of
the Feigenbaum Diagrams
College Mathematics Journal, May 1998, pp. 189-198
Many of us regard the Cantor set as a confusing counterexample to common sense, devised to confound beginning analysis students. Many of us, too, have been introduced to chaos theory and the bifurcation diagrams named for Mitchell Feigenbaum (MR 58 18601). The authors explore beyond the traditional interval, into the region of divergence, and show how Cantor sets arise quite naturally. They do this in a way that can be immediately visualized and comprehended by first-year students -- welcome to Cantors comet!
Aaron Klebanoff is an Associate Professor of Mathematics at Rose-Hulman Institute of Technology. He received his B.S. in mathematics in 1987, his M.S. in applied mathematics in 1989, and his Ph.D. in applied mathematics in 1992 all from the University of California at Davis. His mathematical interests are primarily in fractal geometry and chaotic dynamical systems.
John Rickert is also an Associate Professor of Mathematics at Rose-Hulman Institute of Technology. He earned his B.S. degree in astronomy- physics and mathematics from the University of Wisconsin in 1984, and his Ph.D. in mathematics under the direction of David Masser at the University of Michigan in 1990. His primary mathematical interests are in number theory. His interest in fractals was piqued by a colloquium talk given by Aaron Klebanoff.
Response from Aaron Klebanoff and John Rickert
We are very honored to be recipients of the George Pólya Award. We constantly search for ways to help students visualize complex ideas, so we were touched that our work was judged worthy of this prestigious award. Our thanks go to Robert Devaney for suggesting that we include the basins of attraction in our figures. For the quadratic map, the basin of attraction became affectionately known as the head in Cantors Comet. We also thank Bart Braden and Ellen Curtin for all of their editorial help.