# Richard P Stanley Awards

Richard Stanley has been awarded an undergraduate research prize and four major prizes which we list below giving information about each.

**Click on a link below to go to that award****1. Eric Temple Bell Undergraduate Mathematics Research Prize.**

**1.1. The Eric Temple Bell Undergraduate Prize.**

In 1963 the Department of Mathematics at the California Institute of Technology established the Eric Temple Bell Undergraduate Mathematics Research Prize to honour the memory of Professor Eric Temple Bell and his illustrious career as a research mathematician, teacher, author, and scholar. It is awarded to one or more juniors or seniors for the best original research paper in mathematics. When submitting a paper, students should specify a faculty member who can attest to the quality of the work. The awards committee may award duplicate prizes in case of more than one outstanding entry.

**1.2. The 1965 Bell Prize.**

The 1965 Eric Temple Bell Undergraduate Mathematics Research Prize was awarded to both Michael Ashbacher and Richard P Stanley for outstanding entries. Stanley submitted the paper

*Zero Square Rings*specifying Richard Albert Dean (1924-2022) as a faculty member who could attest to its quality. The paper was published in the Pacific Journal of Mathematics 30 (3) (1969), 811-824. It was the note:

Received September 9, 1968. This paper was written for the 1965 Bell prize at the California Institute of Technology, under the guidance of Professor Richard A Dean.

**1.3. Abstract of Richard P Stanley's**

*Zero Square Rings.*A ring $R$ for which $x^{2} = 0$ for all $x \in R$ is called a zero square ring. Zero-square rings are easily seen to be locally nilpotent. This leads to two problems: (1) constructing finitely generated zero-square rings with large index of nilpotence, and (2) investigating the structure of finitely generated zero square rings with given index of nilpotence. For the first problem we construct a class of zero-square rings, called free zero-square rings, whose index of nilpotence can be arbitrarily large. We show that every zero-square ring whose generators have (additive) orders dividing the orders of the generators of some free zero-square ring is a homomorphic image of the free ring. For the second problem, we assume $R^{n} ≠ 0$ and obtain conditions on the additive group $R_{+}$ of $R$ (and thus also on the order of $R$). When $n = 2$, we completely characterise $R_{+}$. When $n > 3$ we obtain the smallest possible number of generators of $R_{+}$, and the smallest number of generators of order 2 in a minimal set of generators. We also determine the possible orders of $R$.

**2. SIAM George Pólya Prize in Applied Combinatorics (1975).**

**2.1. George Pólya Prize in Applied Combinatorics.**

The George Pólya Prize in Applied Combinatorics, originally established in 1969, is awarded by the Society for Industrial and Applied Mathematics (SIAM) every four years for a notable application of combinatorial theory. The prize is broadly intended to recognise specific work. The award may occasionally be made for cumulative work, but such awards should be rare.

The prize committee will consist of a panel of five SIAM members appointed by the president. One of the members will be designated by the president as chair.

The recipient of the George Pólya Prize in Applied Combinatorics shall receive an engraved medal, a cash award, and reasonable expenses for travel to the award ceremony from the prize fund. The amount of the award is $10,000.

In the exceptional case of multiple winners, the prize money will be shared and reasonable travel expenses for one recipient to attend the award ceremony will be provided.

**2.2. Stanley wins George Pólya Prize in Applied Combinatorics.**

Richard Stanley writes in

*Enumerative and Algebraic Combinatorics in the*

*1960's and 1970's*:-

The first prize to be established in combinatorics was the George Pólya Prize in Applied Combinatorics, awarded by the Society for Industrial and Applied Mathematics (SIAM). The recipients prior to 1980 were Ronald Graham, Klaus Leeb, Bruce Rothschild, Alfred Hales, and Robert Jewett in 1971, Richard Stanley, Endre Szemerédi, and Richard Wilson in 1975, and László Lovász in 1979. The 1975 prize was awarded in San Francisco, so as a bonus Richard Wilson and I (Szemerédi was not present) were invited by Pólya, then aged 87, to visit his home in Palo Alto. We spent an unforgettable evening being shown by Pólya his scrapbooks and other memorabilia.

**3. Leroy P Steele Prize for Mathematical Exposition (2001).**

**3.1. 2001 Steele Prize.**

The 2001 Leroy P Steele Prizes were awarded at the 107th Annual Meeting of the AMS in January 2001 in New Orleans.

The Steele Prizes were established in 1970 in honour of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein and are endowed under the terms of a bequest from Leroy P Steele. The prizes are awarded in three categories: for expository writing, for a research paper of fundamental and lasting importance, and for cumulative influence extending over a career. The current award is $4,000 in each category (in case of multiple recipients, the amount is divided equally).

The recipients of the 2001 Steele Prizes are Richard P Stanley for Mathematical Exposition ...

The Steele Prizes are awarded by the AMS Council acting through a selection committee whose members at the time of these selections were: Constantine M Dafermos, Bertram Kostant, Hugh L Montgomery, Marc A Rieffel, Jonathan M Rosenberg, Barry Simon, François Treves (chair), S R S Varadhan, and Herbert S Wilf.

**3.2. AMS Steele Prize for Mathematical Exposition: Press Release.**

Richard P Stanley, a mathematician at the Massachusetts Institute of Technology, has won the 2001 Leroy P Steele Prize for Mathematical Exposition. Presented by the American Mathematical Society, the Steele Prize is one of the highest distinctions in mathematics. The prize will be awarded today (11 January 2001) at the Joint Mathematics Meetings in New Orleans.

Professor Stanley is being honoured for his two-volume work

*Enumerative Combinatorics*(volume 1 published by Wadsworth & Brooks/Cole in 1986, and volume 2 published by Cambridge University Press in 1999). The citation for the prize says that the field of enumerative combinatorics "has been expanding and evolving very rapidly, and it is quite remarkable that Stanley has been able to take a still photograph of it, so to speak, that beautifully captures its subject."

**3.3. Leroy P Steele Prize for Mathematical Exposition: Richard P Stanley. Citation.**

The Leroy P Steele Prize for Mathematical Exposition is awarded to Richard P Stanley of the Massachusetts Institute of Technology in recognition of the completion of his two-volume work

*Enumerative Combinatorics*. The first volume appeared in 1986, and, to quote the review of Volume 2 by Ira Gessel, "since then, its readers have eagerly awaited Volume 2. They will not be disappointed. Volume 2 not only lives up to the high standards set by Volume 1, but surpasses them. The text gives an excellent account of the basic topics of enumerative combinatorics not covered in Volume 1, and the exercises cover an enormous amount of additional material."

The field has been expanding and evolving very rapidly, and it is quite remarkable that Stanley has been able to take a still photograph of it, so to speak, that beautifully captures its subject. To appreciate the scholarly qualities of this work, one need look no further than the exercises. There are roughly 250 exercises in each volume, all graded according to difficulty, many being multipart, and all with solutions and/or references to the relevant literature being provided. There are more than 500 bibliographic citations in the two volumes.

The first volume begins with elementary counting methods, such as the sieve method, and works through the theory of partially ordered sets, ending with a beautiful treatment of rational generating functions. Volume 2 begins with an advanced, yet very clear, view of generating functions, with special attention to algebraic and D-finite ones, and concludes with a comprehensive discussion of symmetric functions.

Yet even with all of the information that is being transmitted, we never lose clarity or our view of "the big picture". As a small example, we note that the Catalan numbers seem ubiquitous in combinatorics. Every student of the subject is struck by the large number of questions that they answer and wonders if there are bijections between the various families of objects that are counted by these numbers. In a single exercise (ex. 6.19) Stanley has collected 66 such questions and asks the reader to provide the proofs which, in each case, establish the Catalan answer. All 66 of them are worked out in the solution, which is ten pages long, and this is just one of the 500 or so exercises. The author even has time for an occasional smile-generator (e.g., ex. 6.24: "Explain the following sequence: un, dos, tres, quatre, cine, sis,...." The solution tells us that they are the Catalan numbers.)

This is a masterful work of scholarship which is, at the same time, eminently readable and teachable. It will be the standard work in the field for years to come.

**3.4. Richard P Stanley: Biographical Sketch.**

Richard Stanley was born in New York City in 1944. He graduated from Savannah High School in 1962 and Caltech in 1966. He received his Ph.D. from Harvard University in 1971 under the direction of Gian-Carlo Rota. He was a Moore Instructor at Massachusetts Institute of Technology during 1970-71 and a Miller Research Fellow at Berkeley during 1971-73. He then returned to MIT, where he is now a professor of applied mathematics. He is a member of the American Academy of Arts and Sciences and the National Academy of Sciences, and in 1975 he was awarded the Pólya Prize in Applied Combinatorics from the Society for Industrial and Applied Mathematics. His main mathematical interest is combinatorics, especially its connections with such other branches of mathematics as commutative algebra and representation theory.

**3.5. Richard P Stanley: Response.**

I have been interested in expository writing since graduate school and have long admired such masters as Donald Knuth, George Pólya, and Jean-Pierre Serre. I think it is wonderful that the AMS awards a prize for mathematical exposition, and I am extremely pleased at having been chosen for this award. It is not just an award for me but for all of combinatorics, for which such recognition would have been unthinkable when I was starting out in the subject. I only regret that it is not possible for me to share the celebration of my prize with Gian-Carlo Rota, who inspired me throughout my career and who wrote the two forewords to

*Enumerative Combinatorics*.

**4. Rolf Schock Prize in Mathematics (2003).**

**4.1. The Rolf Schock Prize.**

Rolf Schock, who died in 1986, specified in his will that half of his estate should be used to fund four prizes in the fields of logic and philosophy, mathematics, the visual arts and the musical arts.

The fortune is managed by the Schock Foundation, which decides on the prizes on the basis of proposals from the Royal Swedish Academy of Sciences (logic and philosophy and mathematics), the Royal Academy of Fine Arts (visual arts) and the Royal Swedish Academy of Music (musical arts).

The Royal Swedish Academy of Sciences appoint prize committees in logic and philosophy and mathematics that send out nomination invitations and then review the nominated candidates. It is not possible to nominate without an invitation from the committee, or to nominate yourself.

**4.2. The 2003 Rolf Schock Prizes**

The Rolf Schock Prizes for 2003 amounting to SEK 1.6 million are awarded to the logician Solomon Feferman, USA, to the mathematician Richard P. Stanley, USA, to the artist Susan Rothenberg, USA and to the mezzo-soprano Anne Sofie von Otter, Sweden. The Rolf Schock Prize in Mathematics is awarded to Richard P Stanley, Massachusetts Institute of Technology, USA:-

... for his fundamental contributions to combinatorics and its relationship to algebra and geometry, in particular for his important contributions to the theory of convex polytopes and his innovative work on enumerative combinatorics..

**4.3. Richard Stanley Receives the Schock Prize.**

Four Rolf Schock Prizes for 2003 have been awarded, two of them to mathematicians: Solomon Feferman and Richard P Stanley.

The versatile philosopher and artist Rolf Schock (1933-1986) describes in his will a prize to be awarded in such widely differing subjects as logic and philosophy, mathematics, the visual arts, and music. The Royal Swedish Academy of Sciences, the Royal Swedish Academy of Fine Arts, and the Royal Swedish Academy of Music have awarded these prizes every other year since 1993. Each prize carries a monetary award of SEK 400,000 (about US$51,400).

Richard P Stanley

The Schock Prize in Mathematics was awarded to Richard P Stanley of the Massachusetts Institute of Technology:-

... for his fundamental contributions to combinatorics and its relationship to algebra and geometry, in particular for his important contributions to the theory of convex polytopes and his innovative work on enumerative combinatorics.Richard P Stanley has made many pioneering contributions to combinatorics. In addition, he has forcefully and with great originality contributed to the discovery of new connections between combinatorics and other areas of mathematics, to great mutual benefit.

Among his most significant results are his contributions to the study of convex polytopes, the bodies that in higher dimensions correspond to three-dimensional polyhedra (such as cubes and pyramids), especially his proof of necessity in the characterisation of $f$-vectors of simplicial polytopes via algebraic geometry (toric varieties). Furthermore, he has produced first-rate work prompted by enumerative problems, which he often solves in unexpected ways using techniques primarily from commutative algebra, algebraic and convex geometry, and representation theory. His ideas have not only influenced and altered combinatorics profoundly and permanently; they have also stimulated research in the other areas mentioned.

Stanley's scientific production is marked by clarity, breadth, substance, and originality. The methods he has introduced are innovative and have led to decisive progress in many areas of mathematics. He has also spent much effort in writing graduate-level textbooks that have rapidly set the norm.

Richard P Stanley was born in New York in 1944. He studied at the California Institute of Technology and Harvard University, where he received his doctorate in 1971. Since 1979 he has been professor of applied mathematics at MIT. He has been a visiting professor at a number of universities in the United States and France, and also at Stockholm University and the Royal Institute of Technology in Stockholm.

Mathematicians who have previously received the Schock Prize are: Elliott H Lieb (2001), Yuri Manin (1999), Dana S Scott (1997), Mikio Sato (1997), Andrew Wiles (1995), and Elias M Stein (1993).

**5. Leroy P Steele Prize for Lifetime Achievement (2022).**

**5.1. About the Award.**

Presented annually, the American Mathematical Society Leroy P Steele Prize for Lifetime Achievement is awarded for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through PhD students. The Steele Prizes were established in 1970 in honour of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein, and are endowed under the terms of a bequest from Leroy P Steele.

The 2022 prize ws presented on Wednesday, 5 January, during the Joint Prize Session at the 2022 Joint Mathematics Meetings in Seattle.

**5.2. Richard P Stanley receives 2022 Steele Prize for Lifetime Achievement.**

Richard P Stanley, an emeritus professor of mathematics at the Massachusetts Institute of Technology, received the 2022 AMS Leroy P Steele Prize for Lifetime Achievement. Stanley has revolutionised enumerative combinatorics, revealing deep connections with other branches of mathematics, such as commutative algebra, topology, algebraic geometry, probability, convex geometry, and representation theory. In doing so, he solved important longstanding combinatorial problems, often reinvigorating these other fields with new combinatorial methods. Through his outstanding research; excellent expository works; and many PhD students, collaborators and colleagues, he continues to influence the field of combinatorics worldwide.

**5.3. 2022 Steele Prize for Lifetime Achievement: Citation.**

Richard P Stanley, an emeritus professor of mathematics at the Massachusetts Institute of Technology, will receive the 2022 AMS Leroy P Steele Prize for Lifetime Achievement.

Richard Stanley has been a giant in combinatorics and related areas for over four decades. He has revolutionised enumerative combinatorics, revealing deep connections with other branches of mathematics, such as commutative algebra, topology, algebraic geometry, probability, convex geometry, and representation theory. In doing so, he solved important longstanding combinatorial problems, often reinvigorating these other fields with new combinatorial methods. His pioneering work transformed enumerative combinatorics from a disparate collection of clever tricks into a well-structured and highly developed central field of modern mathematics. To this day, his enormous influence on the field continues to grow.

Stanley has repeatedly shown that concrete combinatorics problems can give rise to deep underlying theories of broad interest. His major contributions are numerous and widespread, including the following:

**Combinatorial Reciprocity Theorems.**Stanley explained why many enumerative formulas that depend polynomially on a positive integer $k$ still have combinatorial meaning upon replacing $k$ by $−k$ (e.g., counting subsets versus multisets, counting proper graph colourings versus acyclic orientations, and Ehrhart-Macdonald Reciprocity for counting integer points in polytopes).

He connected reciprocity to the theory of Cohen-Macaulay rings, their canonical modules, and local cohomology, leading to new examples involving linear homogeneous Diophantine equations.

**The Upper Bound Conjecture for Spheres.**In a celebrated paper, Stanley proved this longstanding conjecture by introducing tools from commutative algebra, such as Stanley-Reisner rings and Cohen-Macaulay simplicial complexes. These notions evolved into core topics of algebraic, geometric, and topological combinatorics.

**Combinatorial applications of the Hard Lefschetz Theorem.**Stanley introduced into combinatorics the use of the Hard Lefschetz Theorem from algebraic geometry. His major breakthroughs include the proof of the necessity part of McMullen's g-conjecture for convex polytopes, and the proof of a Sperner theorem for certain partially ordered sets that resolved a longstanding number theoretic conjecture of Erdős and Moser. This and other techniques imported by Stanley from algebraic geometry and representation theory of $SL(2)$ are part of the Kähler package, which continues to yield striking combinatorial positivity and unimodality results to this day.

**Application of the theory of symmetric functions.**Stanley recognised that the theory of symmetric functions, and its classical relation to symmetric group representations, had far-reaching applications in combinatorics, for example, to partition identities, permutation statistics, and enumeration of reduced words in Coxeter groups (e.g., the symmetric group). He introduced two classes of symmetric functions, which have generated an explosion of activity in combinatorics and also in algebraic geometry. One now known as the Stanley symmetric function is related to the Schubert calculus, and the other, his chromatic symmetric function, was recently shown to be related to cohomology of Hessenberg varieties.

Stanley is a gifted writer. His award-winning books Enumerative Combinatorics Vol. I and Vol. II are landmarks of inspired and elegant exposition. They are invaluable resources that quickly became bibles in the subject. Their extensive exercises alone have kindled an enormous amount of research in this ever expanding field.

Stanley has supervised 60 MIT and Harvard PhD students and mentored an immense number of postdocs and visitors. Many of these mathematicians and their descendants went on to become influential leaders at top universities.

**5.4. Richard P Stanley: Biographical Note.**

Richard P Stanley attended the California Institute of Technology as an undergraduate and received his PhD from Harvard University in 1971. He originally planned to work in algebra or number theory, but under the spell of Gian-Carlo Rota, he switched his main research interest to combinatorics. After postdocs at MIT and UC Berkeley, Stanley returned in 1973 to MIT, where he remained until retiring in 2018. He continues as an Arts and Sciences Distinguished Professor at the University of Miami during spring semesters. He especially likes connections between combinatorics and other branches of mathematics.

Stanley is a member of the American Academy of Art and Sciences and the National Academy of Sciences and a Fellow of the AMS. He was a plenary speaker at the ICM in 2006 and gave the AMS Colloquium Lectures in 2010. He received the SIAM George Pólya Prize in Combinatorics in 1975, the Steele Prize for Mathematical Exposition in 2001, and the Rolf Schock Prize in Mathematics in 2003. Sixty students received their PhDs under his supervision. In addition to these mathematical progeny, he has two biological children and two grandchildren.

**5.5. Response from Richard P Stanley.**

It is a wonderful honour to receive the Leroy P Steele Prize for Lifetime Achievement. When I started working in enumerative and algebraic combinatorics around 1967, I was greatly attracted to the vision of my thesis adviser Gian-Carlo Rota (recipient of a 1988 Steele Prize), who saw glimmerings of deep connections between combinatorics and other branches of mathematics. In graduate school, I became interested in partially ordered sets and symmetric functions. They fortunately turned out to be extremely fecund concepts which continue to fascinate me. I have described elsewhere how the problem raised by MacMahon of enumerating solid (3-dimensional) partitions eventually led me to find connections with convex polytopes, commutative algebra, and algebraic geometry, leading to the solution of a number of problems, such as the Upper Bound Conjecture for Spheres and the g-conjecture for simplicial polytopes, having nothing to do with solid partitions. It is interesting how the path to successful research can be so circuitous.

Throughout my career it has been gratifying to see many highly talented aspiring mathematicians decide to work in algebraic and enumerative combinatorics. Thanks to their efforts, the field has become vastly more sophisticated and intertwined with other areas than when I began my own research. This Steele Prize should be regarded not only as an individual honour, but also as a testament to the efforts of these other researchers who have raised algebraic and enumerative combinatorics to its present lofty level. I should also express my gratitude to the MIT Department of Mathematics for providing a stimulating and nurturing environment for learning, discovering, and teaching mathematics.

Last Updated March 2024