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August 6, 2012 |

The Mathematical Association of America has selected David A. Cox (Amherst College), Graham Everest (University of East Anglia) and Thomas Ward, Peter Sarnak (Princeton University), and Ravi Vakil (Stanford University) as the 2012 winners of the Lester R. Ford Award. Full citations and biographical information for each winner are available below.

The Lester R. Ford Awards were established in 1964 to recognize authors of articles of expository excellence published in *The American Mathematical Monthly* or*Mathematics Magazine*. Beginning in 1976, a separate award (the Allendoerfer Award) was created for *Mathematics Magazine*. The awards are named for Lester R. Ford, Sr., a distinguished mathematician, editor of the *American Mathematical Monthly,* 1942-1946, and President of the Mathematical Association of America, 1947-1948. This is an award of $500. Up to five of these awards are given annually at the Summer Meeting of the Association. All awards given from 1976 on are for articles that appeared in *The American Mathematical Monthly*. Between 1965 and 1975, Ford awards were given for articles in the *Monthly* or *Mathematics Magazine*.

Awards were presented during the MAA Prize Session on Friday, August 3, 2012, at the 2012 MAA MathFest in Madison, Wisconsin.

?Why Eisenstein Proved the Eisenstein Criterion and Why SchÃ¶nemann Discovered It First,?American Mathematical Monthly, 118:1 (2011), p. 3-21.

We all recall Eisenstein?s criterion mostly for its application to show
that the cyclotomic polynomial for prime *p* is irreducible and perhaps
we wonder why a name is attached to this, seemingly minor, auxiliary
result. In this fascinating paper David Cox not only tells us that
Eisenstein was scooped by Theodor SchÃ¶nemann but, much more
interestingly and importantly, he explains why both men were led to
the result.

It?s an engrossing tale beginning with Gauss?s equal division of the circle, the relation of that work to the analogous problem on a lemniscate, the connection of that problem to the question of solvability by radicals of polynomials, and thence into the wonderlands of Galois theory, finite fields, and elliptic curves. It is an amazingly rich story, beautifully told, not of a priority dispute but of a grand sweeping flow of ideas beginning with Gauss (who partially scooped both SchÃ¶nemann and Eisenstein) and extending into the beating heart of modern-day mathematics. It is a tour de force of mathematical and historical scholarship.

David A. Cox went to Rice University and received his Ph.D. from Princeton University in 1975. After teaching at Haverford and Rutgers, he has been at Amherst College since 1979, except for a sabbatical at Oklahoma State University. After more than 30 years, he still loves the combination of teaching and scholarship that is possible at a liberal arts college. His current areas of research include toric varieties and the commutative algebra of curve parametrizations. His earlier work in algebraic geometry includes papers on Ã©tale homotopy theory, elliptic surfaces, and infinitesimal variations of Hodge structure, and he also has interests in number theory and the history of mathematics. He is the author of books on number theory, computational algebraic geometry, mirror symmetry, Galois theory, and toric varieties, three of which have been translated into Japanese.

?A Repulsion Motif in Diophantine Equations,?American Mathematical Monthly, 118:7 (2011), p. 584-598.

Beginning with *y*^{2} + 2 = *x*^{3}, the authors entice the reader with the
distinguished history of this equation along with the surprising
sizes of solutions. The authors then lead the reader forward in
time, effectively offering a ?speed dating? tour of highlights in
Diophantine equations, such as the abc conjecture, the Baker-Stark
methods, the recent proof of the Catalan conjecture, and the
geometry of elliptic curves. They deftly introduce key definitions
and themes of Diophantine equations in simple concrete contexts,
gently hinting at the complexity that a fully general description
would involve. The authors weave several themes throughout the
article, such as the interplay of computation/conjecture/theory, or
the ?familiar refrain? that an effective (bounded) search may still
be an impracticable one. This paper exemplifies the *Monthly*?s goal
to ?inform, stimulate, challenge, enlighten, and even entertain? its
readers.

**Tom Ward** has worked at the University of East Anglia since 1992,
and is currently Pro-Vice-Chancellor (Academic) with responsibility
for teaching and learning and the student experience. He attended
Waterford-Kamhlaba School in Swaziland, where he encountered
several inspirational mathematics and physics teachers who nurtured
an interest started by his physicist parents. After studying at the
University of Warwick, he worked at College Park and Ohio State
University before returning to England. He works in ergodic theory,
and enjoyed a long collaboration with Graham Everest, studying
dynamical systems from a number-theoretic point of view and
number theory from a dynamical point of view. He has written
several books, including *Recurrence sequences* with Everest, Alf van der Poorten, and Igor Shparlinski, *Heights of polynomials and
entropy in algebraic dynamics* with Everest, and *Ergodic theory
with a view towards number theory* with Manfred Einsiedler.

**Graham Everest**, who was elected a member of the London
Mathematical Society in 1983, died on 30 July 2010, aged 52.

*Thomas Ward writes*: Graham?s talent for mathematics took him to
Bedford College and doctoral study under the supervision of Colin
Bushnell at King?s College London. He joined the University of East
Anglia as a lecturer in 1983, and spent his whole career there.

His research appeared in the form of some 70 research papers and three monographs, and spanned diverse areas of number theory. Three themes informed his research. First, the impact of twentieth-century developments in Diophantine analysis and transcendence theory on counting problems and questions in algebraic number theory. Second, the fascinating arithmetic properties of recurrence sequences, including classical questions in the spirit of Mersenne, Lehmer, Zsigmondy, and so on, as well as more modern developments on bilinear sequences and elliptic divisibility sequences. Third, Graham had an abiding interest in all aspects of the interaction between number theory and dynamical systems.

As a researcher Graham brought great joy and creativity to his work, and the generosity of his approach to mathematics will be familiar to his thirty co-authors. Graham was a dedicated teacher and supervisor, and many generations of students will remember the energy and enthusiasm of his lectures. His belief in the transforming power of higher education was recognized in the form of a UEA Excellence in Teaching award in 2005.

?Integral Apollonian Packings,?American Mathematical Monthly, 118:4 (2011), p. 291-306.

This wonderful paper begins by considering three coins’a nickel, a dime, and a quarter. A theorem of Apollonius says that another coin can be placed in the region that they bound so that all four coins are mutually tangent. Actually, Apollonius?s theorem says more: given any three mutually tangent circles, there are two circles tangent to all three. This paper is about the radii of these circles, investigated through the curvature (reciprocal of the radius). Descartes established a beautiful relation among the five curvatures, and his result implies that the radii of all further circles lie in an extension field of the rationals (it is generated by just one square root obtained from the three original radii).

Another consequence of Descartes? result is that when the three original curvatures are all integers and one other elementary condition is satisfied, then all of the subsequent curvatures are integers. This is the point at which this article takes off ? it leads to connections with several other areas of mathematics, and what the author does so marvelously is to acquaint the reader with several of these. They include algebra through the Apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime.

In his exposition, the author has skillfully combined content, old with new, elementary with advanced. From its humble beginnings of three coins and the results of Apollonius and Descartes through to fascinating recent results and open problems, ?Integral Apollonian Packings? is truly an exceptional article.

**Peter Sarnak** is a Professor of Mathematics at Princeton University
and the Institute for Advanced Study, Princeton. He received a B.S.
degree from the University of Witwatersrand (Johannesburg) and a
Ph.D. from Stanford University. His mathematical interests are wide-ranging and his research focuses on problems in number theory,
automorphic forms, geometric analysis and related combinatorics,
and mathematical physics.

?The Mathematics of Doodling,?American Mathematical Monthly, 118:2 (2011), p. 116-129.

This article, based on the first of Vakil?s Hedrick lectures at the 2009
MathFest, is a wonderful example of expository mathematics. A
*doodle* involves starting with a shape (for example a ?W?) on a piece
of paper, and then drawing a curve around it, roughly the set of
points within a small constant "distance" from the W. Now repeat
the procedure starting with the curve obtained and keep repeating.
Do the successive doodles get more and more "circular?" The
author began with this simple mathematical question, one that a
seventh grader might ask, and "just followed where it took us."

Definitions and questions are made more precise, and eventually lead to reworded precise and satisfying answers. Just following ?where it took us? inspires a sequence of natural generalizations and, inevitably, to more sophisticated topics. Vakil touches on the relevance of these investigations to elementary, and not so elementary, well-known and not so well-known problems in geometry. Along the way the reader gets an informal introduction to linear invariants, winding numbers, differential geometry, Hilbert?s third problem, and current research in algebraic and hyperbolic geometry. As Vakil concludes: "In some sense our journey is a metaphor for mathematical exploration in general."

**Ravi Vakil** is a Professor of Mathematics and the Packard University
Fellow at Stanford. He is an algebraic geometer, whose work touches
on topology, string theory, applied mathematics, combinatorics,
number theory, and more. He was a four-time Putnam Fellow while
at the University of Toronto. He received his Ph.D. from Harvard,
and taught at Princeton and MIT before moving to Stanford. He
has received the Dean's Award for Distinguished Teaching, the
American Mathematical Society Centennial Fellowship, the Terman
fellowship, a Sloan Research Fellowship, the NSF CAREER grant,
and the Presidential Early Career Award for Scientists and Engineers.
He has also received the Coxeter-James Prize from the Canadian
Mathematical Society, and AndrÃ©-Aisenstadt Prize. He was the 2009
Hedrick Lecturer at MathFest, and is a MAA PÃ³lya Lecturer. He is
an informal advisor to the website mathoverflow. He works extensively
with talented younger mathematicians at all levels, from high school
through recent Ph.D.'s.