Invited Speakers

Invited Speakers for Supplemental Lectures & Tutorials:

George Andrews (Pennsylvania State University)

Madeline Dawsey (University of Texas Tyler)

Christopher Jennings-Shaffer (Fred Hutch Cancer Research Center)

Peter Paule (Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria)

Larry Rolen (Vanderbilt University)

Ae Ja Yee (Penn State University)

Speaker: George Andrews, Pennsylvania State University
Title: Partition Identities for k-Regular Partitions with Distinct Parts

Abstract: We start with a little-known Euler type theorem (due to Alladi) which is the following:

The number of partitions of n into distinct parts not divisible by k (i.e. k-regular partitions with distinct parts) equals the number of partitions of n into odd parts none repeated more than k-1 times. k=1 and 2 are tautologies. k=3 plays a prominent role in Schur's 1926 partition theorem. Both Alladi and Schur have further partition identities related to k=2 which we will discuss. Obviously, k = infinity is Euler's theorem. We then proceed to k=4 where an empirical investigation leads to a result for overpartitions. We conclude with a proof of the k=4 case and look at results and possibilities for k>4.

Speaker: Madeline Dawsey, University of Texas Tyler

Title: Ranks, Cranks, and Partition Congruences

Description: Three partition congruences were discovered by Ramanujan in 1919, and since then infinitely many congruences have been found by various mathematicians. Ramanujan’s congruences are still (provably) the simplest, most beautiful examples in partition theory. We will see many examples of partition congruences and related results, and we will learn how they led to Dyson’s famous conjecture of the existence of a partition statistic which combinatorially explains Ramanujan’s three congruences. Dyson defined the rank of a partition, which partially proves his conjecture. We will experiment with ranks, see why they fail to prove the third of Ramanujan’s congruences, and then explore a different partition statistic called the crank which successfully explains all three.

Speaker: Christopher Jennings-Shaffer, Fred Hutch Cancer Research Center

Title: TBA

Speaker: Peter Paule, Research Institute for Symbolic Computation, Johannes Kepler University Linz (research talk)

Title: Differential Equations and Modular Forms

Abstract: The study of holonomic functions and sequences satisfying linear

differential and difference equations, respectively, with polynomial

coefficients has roots tracing back to the time of Gauss (at least).

Tools to assist this study, including methods from computer algebra,

have become fundamental for the modern theory of enumerative

combinatorics; see, e.g., the work of Stanley and Zeilberger.

Also tracing back to the time of Gauss (at least) are highly

non-holonomic objects: modular functions and modular forms with

q-series representations arising also as generating functions of

partitions of various kinds. Using computer algebra, the talk

connects these two different worlds. Applications concern partition

congruences, Fricke–Klein relations, irrationality proofs a la

Beukers, or approximations to pi studied by Ramanujan and the

Borweins. As a major ingredient to a "first guess, then prove"

strategy, a new algorithm for proving differential equations

for modular forms is used. The results presented arose in

joint work with Silviu Radu (RISC).

Speaker: Peter Paule, Research Institute for Symbolic Computation, Johannes Kepler University Linz (software talk)

Title: RISC Computer Algebra Packages for q-Series

Abstract: This expository talk introduces to computer

algebra packages developed in my group at RISC. Besides

sketching underlying theory, special emphasis is put on

illustrative examples. Applications concern q-hypergeometric

sums and series, q-holonomic functions and sequences, as

well as tools related to modular functions.

Speaker: Larry Rolen, Vanderbilt University

Title: Recent problems in partitions and other combinatorial functions

Abstract: In this talk, I will discuss recent work, joint with a number of

collaborators, on analytic and combinatorial properties of the partition

and related functions. This includes work on recent conjectures of Stanton,

which aim to give a deeper understanding into the "rank" and "crank"

functions which "explain" the famous partition congruences of Ramanujan. I

will describe progress in producing such functions for other combinatorial

functions using the theory of modular and Jacobi forms and recent

connections with Lie-theoretic objects due to Gritsenko--Skoruppa--Zagier. I

will also discuss how analytic questions about partitions can be used to

study Stanton's conjectures, as well as recent conjectures on partition

inequalities due to Chern--Fu--Tang and Heim--Neuhauser, which are related to

the Nekrasov--Okounkov formula.

Speaker: Ae Ja Yee, Pennsylvania State University

Title: Partition rank and crank from a combinatorial point of view

Abstract: Integer partitions carry various interesting statistics. Of those, partition ranks and cranks are the most loved and studied ones. In 1944, Freeman Dyson defined rank statistic claiming that it combinatorially explains Ramanujan's mod 5 and 7 partition congruences. Dyson's claim was confirmed by Atkin and Swinnerton-Dyer in 1955. In the same paper, Dyson also conjectured the existence of another statistic for the mod 11 congruence, namely crank, and this conjecture was settled by Andrews and Garvan in 1988. Since then, rank and crank have received a lot of attention. In this lecture, I will survey results on partition rank and crank presenting their significances in the theory of partitions.