2006 HMC Mathematics Conference on Enumerative Combinatorics: Abstracts
George E. Andrews (Pennsylvania State University): Partition Analysis and the Search for Modular Forms
This talk will introduce P.A. MacMahon's method of partition analysis. We will the discuss some of its applications including our most recent discovery of certain partitions realted to directed graphs that have modular forms as generating functions and consequently have a variety of interesting related arithmetical problems and prospects. This work is joint with Peter Paule and Axel Riese.
Ken Ono (University of Wisconsin): Freeman Dyson's Challenge for the Future: The Story of Ramanujan's Mock Theta Functions
The legend of Ramanujan is one of the most romantic stories in the modern history of mathematics. It is the story of an untrained mathematician, from south India, who brilliantly discovers tantalizing examples of phenomena well before their time. Indeed, the legacy of Ramanujan's work (as a whole) is well documented and includes direct connections to some of the deepest results in modern number theory such as the proof of the Weil Conjectures, and the proof of Fermat's Last Theorem. However, one final problem remained. In his last letter to Hardy (written on his death bed), Ramanujan gave examples of 17 functions he referred to as "mock theta functions". Without a definition and without good clues, number theorists were unable to make any real sense out of these peculiar functions. Nevertheless, these examples make important appearances in many disparate areas of mathematics, a fact which inspired Freeman Dyson to proclaim:
Mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure... This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions.
—Freeman Dyson, 1987
In this lecture I will outline the history of Ramanujan's final enigma, and I will announce the solution.
Igor Pak (MIT): The Nature of Partition Bijections
The study of partition identities has a long history going back to Euler, with applications ranging from analysis to number theory, from enumerative combinatorics to probability. Partition bijections is a combinatorial approach which often gives the shortest and the most elegant proofs of these identities. These bijections are then often used to generalize the identities, find “hidden symmetries”, etc.
In the talk I will present a number of partition bijections and discuss their applications. The talk assumes no background whatsoever, and hopefully will be somewhat entertaining.
Igor Pak (MIT): MacMahon's Master Theorem
MacMahon's Master Theorem is a classical combinatorial result celebrated by its applications to binomial identities. In this talk I will give an introduction to the subject, present an algebraic and a direct bijective proof of the theorem. If time permits, I will also discuss various extensions of the Master Theorem.
Carla D. Savage (North Carolina State University): The Binomial Essence of Lecture Hall Partitions
A lecture hall partition is a sequence x1, x2, ... , xn of nonnegative integers satisfying x1/n ≥ x2/(n-1) ≥ ... ≥ xn/1. Lecture hall partitions were introduced in 1997 by Bousquet-Mélou and Eriksson who showed that they are in one-to-one correspondence with partitions into odd parts less than 2n. Since then, several generalizations and refinements of this result have been discovered. In this talk we view lecture hall partitions from three different perspectives (integer-analogs, q-series identities, and Sylvester's bijection) to uncover some new connections.
Doron Zeilberger (Rutgers University): Beautiful and Insightful Computer-Generated Bijective Proofs
Computer-Generated proofs do not have to be artificial, and many human-made proofs are not that natural. It is possible to take “artificial” proofs and naturalize them, and we can even cheat and pretend that we did it all by ourselves. (Joint work with Philip Matchett Wood.)