Combinatorics via modular forms | Department of Mathematics

Combinatorics via modular forms

Event Information
Event Location: 
GAB 317, 4-5 PM; Refreshments: GAB 472, 3:30 PM
Event Date: 
Monday, February 21, 2011 - 4:00pm

We show how to answer some fundamental questions in combinatorial number theory using modular forms. Modular forms are analytic functions which play a central role in modern number theory. We describe a beautiful application to the theory of partitions. Ramanujan famously proved congruences modulo 5, 7, and 11 for the partition-counting function (for example, he showed that p(5n+4) = 0 modulo 5). He speculated that there were no other such congruences, and in 2003 Ahlgren and Boylan proved that this was indeed the case. We provide a broad generalization of this phenomenon. We illustrate with several examples and place this phenomenon in context by giving the exact probability of having a "Ramanujan Congruence".