Talk: A gumbo with hints of partitions, modular forms, special integer sequences and supercongruences (Illinois)

Date: 2017/03/16
Occasion: Number Theory Seminar
Place: University of Illinois at Urbana-Champaign

Abstract

Euler's partition theorem famously asserts that the number of ways to partition an integer into distinct parts is the same as the number of ways to partition it into odd parts. In the first part of this talk, we describe a new analog of this theorem for partitions of fixed perimeter. More generally, we discuss enumeration results for simultaneous core partitions, which originates with an elegant result due to Anderson that the number of \((s,t)\)-core partitions is finite and is given by generalized Catalan numbers.

The second part is concerned with congruences between truncated hypergeometric series and modular forms. Specifically, we discuss a supercongruence modulo \(p^3\) between the \(p\)th Fourier coefficient of a weight 6 modular form and a truncated \(_6F_5\)-hypergeometric series. The story is intimately tied with Apéry's proof of the irrationality of \(\zeta(3)\). This is recent joint work with Robert Osburn and Wadim Zudilin.

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