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September 8, 2020

Partition congruences and the localization method

A notable problem in partition theory is the study of infinite families of partition congruences modulo powers of a prime. It has recently been discovered that there exist congruence families, associated with a modular curve of genus 0, for which the traditional methods of proof fail. One such congruence family is related to the spt analogue of the omega mock theta function. We recently gave a proof of this congruence family by a new method, based on the manipulation of a localized polynomial ring, rather than by studying Z[X] via the more classical methods. We will give a brief outline of this method, its surprisingly unique characteristics, and its potential for future work.