Modular Generating Functions: A powerful device | Department of Mathematics

Modular Generating Functions: A powerful device

Event Information
Event Location: 
GAB 461, 4-5 PM; Refreshments: GAB 472, 3:30 PM
Event Date: 
Monday, March 3, 2014 - 4:00pm

We revisit the concept of generating functions. Multinomial polynomials and Schur polynomials are two of the most classical instances. We then start discussion of modular generating functions by enumerating some that appear in areas related to the faculty's fields of interest. Based on this list, we discuss what additional insight can be derived after identifying a generating function as being modular.

After this we properly define the most basic type of modular form. This, already, suffices to formulate the famous Taniyama-Shimura-Weil conjecture, proved by Wiles et al. roughly two decades ago. We highlight differences with more general, modern concepts of modular forms, which can be applied to, e.g., integer partitions and branched covers of Riemann surfaces.

We conclude the talk by providing a perspective on very recent developments, and expected future results.