Event Type:

Department Colloquium

Date/Time:

Thursday, March 3, 2005 - 08:00

Location:

Kidd 364 (<b>Note: unusual day and time</b>)

Guest Speaker:

Institution:

University of Wisconsin

Abstract:

Notice that 4=3+1=2+2=2+1+1=1+1+1+1, and so p(4)=5. At first glance, partitions seem like child's play. However, the study of partitions has a long and storied history highlighted by works of Ramanujan addressing the following questions:

Question 1: What is the size of p(n)? Question 2: Does p(n) have congruence properties?

The study of these questions has led to interesting research in many areas, including number theory, combinatorics, representation theory, and arithmetic geometry. For example, the first question led to the creation of the Hardy-Littlewood-Ramanujan ``circle method" which gave an incredibly accurate asymptotic formula for p(n). The circle method is a powerful tool in analytic number theory, having implications to Waring's problem, Borcherd's theory and other problems. The study of the second question has played a central role in the development of the theory of p-adic modular forms, representations of the symmetric group, and has even led to information about primes dividing the order of certain Shafarevich-Tate groups.

Recently, R. Stanley formulated a new partition function t(n) in his work on sign-balanced posets. Motivated by Stanley's results, G. E. Andrews proved analogues of early works of Ramanujan for t(n). In an address to the AMS in May 2003, Andrews discussed this work and asked the following familiar questions for t(n).

Question 1: What is the size of t(n)? Question 2: Does t(n) have congruence properties?

Here we answer these questions, and show more.