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2016 David P. Robbins Prize 2016 David P. Robbins Prize Christoph Koutschan Manuel Kauers Doron Zeilberger Christoph Koutschan, Manuel Kauers, and symmetry classes of plane partitions were enu- Doron Zeilberger were awarded the 2016 David merated by a family of remarkably simple product P. Robbins Prize at the 122nd Annual Meeting of formulas. In some cases, this was also found to be the AMS in Seattle, Washington, in January 2016 true of the q-analogs of these formulas—generat- for their paper, “Proof of George Andrews’s and ing functions where q tracks some parameter of David Robbins’s q-TSPP conjecture,” Proceedings interest. Simple combinatorial formulas usually of the National Academy of Sciences of the USA, have simple combinatorial explanations, but sur- 108 (2011), 2196–2199. prisingly that is not the case here. Many of the Citation formulas are exceedingly difficult to prove, and to A plane partition π may be thought of as a finite this day there is no unified proof that explains why set of triples of natural numbers with the property they are all so simple and so similar. When simple that if (i, j,k) is in π, then so is every (i,j,k) with combinatorial formulas fail to have simple combi- i≤i, j≤j, and k≤ k. The symmetric group S 3 acts natorial proofs, it usually means that something on plane partitions by permuting the coordinate deep is going on beneath the surface. Indeed, the axes. A plane partition π may also be “comple- study of symmetry classes of plane partitions has mented” by constructing the set of points not in π revealed unexpected connections between several that are in the smallest box enclosing π, and then areas, including statistical mechanics, the repre- reflecting them through the center of the box. The sentation theory of quantum groups, alternating set of plane partitions that is invariant under some sign matrices, and lozenge tilings. subgroup of these transformations is known as a By 1995 all the formulas had been proved but “symmetry class of plane partitions.” one: the q-TSPP (totally symmetric plane parti- In the early 1980s, the combined experimen- tal observations of several people, including tions, where q tracks the number of orbits of David Robbins himself, suggested that all the triples) formula, independently conjectured by George Andrews and David Robbins. The paper For permission to reprint this article, please contact: by Koutschan, Kauers, and Zeilberger, along with [email protected]. supporting computer files on Koutschan’s web- DOI: http://dx.doi.org/10.1090/noti1356 site, finally established the correctness of this 432 NOTICES OF THE AMS VOLUME 63, NUMBER 4 last conjectured formula. It is a tour de force of constantly called “the holy grail of enumerative experimental mathematics. combinatorics.” Back then I was a PhD student and The authors’ starting point was a 1989 paper of quite excited to work on such an old and important Okada that reduced the problem to a conjecture conjecture. At the very beginning I didn’t know at on the values of the determinants of a special all what a totally symmetric plane partition was or sequence of matrices. The form of the matrices what q-enumeration meant. But thanks to Soichi enables one to verify the conjecture by Zeilberger’s Okada, who had reformulated the conjecture in “holonomic ansatz,” in which one hypothesizes a terms of a family of determinants, that wasn’t even system of auxiliary functions that would certify necessary; the only thing one had to do was to eval- the correctness of the determinant evaluation. uate these determinants. And Doron had already If these auxiliary functions exist, then one can devised a recipe—the holonomic ansatz—how one search for them computationally. Once found, could do that, but the necessary computer algebra the functions must be shown to satisfy certain calculations seemed to be far beyond what was recurrences in order to complete the proof. The possible with the current hardware and software. authors performed the search, found the candidate As a warmup example we applied Doron’s recipe auxiliary functions, and empirically verified that to the q=1 case, first proved by John Stembridge in they seemed to satisfy the necessary recurrences. 1995 using a different approach, but already this Although techniques existed that might in princi- considerably easier task appeared to be hopeless. ple find proofs for the recurrences, the necessary At this point, the visionary suggestion of my PhD computations seemed far beyond the reach of advisor Peter Paule to investigate and implement current computers, and the authors initially an- the holonomic systems approach turned out to be nounced a semirigorous but not fully rigorous a lucky chance: by the end of my PhD I had found proof. However, the authors then developed “cre- a way to solve the q=1 case, and, shortly after, a ative telescoping” techniques that dramatically new method for computing creative telescoping simplified the necessary computations, resulting relations was born, which was efficient enough to in a fully rigorous proof of the q-TSPP formula. prove the determinant evaluations that implied The David P. Robbins Prize is awarded to a novel the q-TSPP formula. Since that time this method, research paper in algebra, combinatorics, or dis- which was originally motivated from David Rob- crete mathematics with a significant experimental bins’s conjecture, has been successfully applied to component. The q-TSPP paper is a shining exam- various other problems ranging from enumerative ple. The conjecture itself was born from experi- combinatorics and statistical physics to numerical mental observations, and the proof involved the analysis and knot theory. development of new experimental mathematical Biographical Sketch of Kauers techniques that are sure to solve many problems. Manuel Kauers was born in Germany and studied Biographical Sketch of Koutschan computer science at the University of Karlsruhe Christoph Koutschan, born in 1978, received his from 1998 to 2002 before he went to Linz, Austria, master’s degree in 2005 from Friedrich Alexander to do a PhD in symbolic computation under the University and his PhD in 2009 from the Johannes supervision of Peter Paule. He earned his doctoral Kepler University under the direction of Peter degree in 2005 and an habilitation in mathematics Paule. He was a postdoctoral fellow at the Research in 2008. In 2009, he was awarded the START prize Institute for Symbolic Computation (RISC, Linz), of the Austrian federal minister for science and at Tulane University, and at the Institut National education, the most prestigious award for young de Recherche en Informatique et en Automatique scientists in Austria, endowed with a research (INRIA). He is currently with the Johann Radon In- grant of 1.2 million euros. In June 2015, Kauers was stitute for Computational and Applied Mathemat- appointed full professor at the Johannes Kepler ics (RICAM) of the Austrian Academy of Sciences. University in Linz, where he is now the director His research interests are devoted to computer of the Institute for Algebra. He is best known for algebra methods related to the holonomic systems his contributions to computer algebra and its approach, particularly symbolic summation and applications to combinatorics and experimental integration algorithms, and their application in mathematics. other areas of mathematics. Response from Manuel Kauers Response from Christof Koutschan Back in 2006, Doron Zeilberger visited Linz and I’m deeply honored to receive the David P. Robbins sketched a way to prove the q-TSPP conjecture Prize for the proof of the q-enumeration formula computationally. The computations he suggested for totally symmetric plane partitions. This beauti- were quite expensive, and there was no reason ful and mysterious formula was conjectured inde- (other than his intuition) that the “certificate” he pendently by David Robbins and George Andrews aimed to compute would even exist. We were re- in the early 1980s. It was none other than Doron luctant to invest time and energy on such a risky Zeilberger who proposed to my colleague Manuel project, but Zeilberger kept pushing us until we Kauers and me to tackle this problem, which he did. Now in 2016, I am glad that he insisted, and APRIL 2016 NOTICES OF THE AMS 433 I am happy to receive the David P. Robbins Prize of the problem and clear exposition of the work. together with Koutschan and Zeilberger for a hard The US$5,000 prize is awarded every three years. piece of work that we have done together. I also ap- The David P. Robbins Prize is awarded by the preciate very much that the committee has decided AMS Council acting on the recommendation of a to recognize a paper with a proof that was found selection committee. The members of the 2016 by computer algebra, thereby helping to raise the David P. Robbins Prize Committee were the fol- awareness of the power of today’s computational lowing individuals. methods, not just for industrial applications but •Sara C. Billey also for pure mathematics. •Timothy Y. Chow Biographical Sketch of Zeilberger •Curtis Greene Doron Zeilberger was born on July 2, 1950, in Haifa, •Victor Reiner Israel, to Ruth (Alexander) and Yehudah Zeilberger. •Daniel A. Spielman (Chair) He received his PhD in 1976 from the Weizmann The complete list of recipients of the David P. Institute of Science (as a student of Harry Dym (a Robbins Prize follows. student of Henry McKean (a student of William 2007 Samuel P. Ferguson, Thomas C. Hales Feller (a student of Richard Courant (a student 2010 Ileana Streinu of David Hilbert))))). He is currently the Board 2013 Alexander Razborov of Governors Professor at Rutgers University. In 2016 Christoph Koutschan, Manuel Kauers, 1979 he married Jane D.
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