Professor Olav Richter invites you to attend the

Master's Defense of Jacob Williams

"Ramanujan Congruences for the Partition Function"

ABSTRACT:

The partition function is defined as the number of ways to write a positive integer as a non-increasing sum of positive integers. Ramanujan showed that the partition function satisfies three simple congruences modulo the primes 5, 7, and 11, now called Ramanujan congruences. He also conjectured that no other primes give equally simple congruences. In their celebrated paper published in 2003 in Inventiones Mathematicae, Ahlgren and Boylan proved this conjecture to be true! In this talk, I will review some tools from the theory of modular forms modulo primes, and use these tools to sketch the proof of Ahlgren and Boylan's theorem.