Associate Professor of Mathematics, William Keith, PhD, Michigan Tech University, will present "Ramanujan-Kolberg identities, regular partitions, and multipartitions" on March 8.

Abstract: Ramanujan-Kolberg identities, so named after Ramanujan's "most beautiful identity" $\sum_{m=0}^\infinity p(5m+4) q^n = 5 \prod_{n=1}^\infty \frac{(1-q^{5n})^5}{(1-q^n)^6}$ relate subprogressions of the partition numbers and linear combinations of eta-quotients.

Descending from equality to congruence mod 2, recent work of the speaker and Fabrizio Zanello has produced a large number of these with implications for the study of the parity of the partition function, and Shi-Chao Chen has shown that these are part of an infinite family where the eta-quotients required have a very small basis.

Analyses of particular cases from that family yield many pleasing patterns: older work of the speaker and Zanello gave congruences for the $m$-regular partitions for $m$ odd, and this talk will be on more recent joint work which exhibits the very different behavior for $m$ even which gives connections to multipartitions. All of these results in turn illuminate different aspects of the longstanding partition parity problem and hopefully provide some useful insight therein.