PRIZES and AWARDS
Providence, Rhode Island
Sunday, August 1, 1999
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
Decomposition Theorem
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.
Biographical Notes
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.
Biographical Notes
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.
Ravi Vakil
The Youngest Tenured Professor in Harvard
History
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.
Biographical Note
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.
Yoav Benyamini
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.
Biographical Note
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.
Biographical Notes
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
Bernd Sturmfels
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 +
a4y = b1 +
b2x2y +
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.
Biographical Note
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.
Aleksandar Jurisic
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.
Biographical Note
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.
Biographical Notes
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!
Biographical Notes
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.