DMS Colloquium: Dr. Matthew R. Ballard

Speaker:   Dr. Matthew R. Ballard (University of South Carolina)

Title: How complex are modules over quotients of polynomial rings?

Abstract: From linear algebra, we know that a single number captures (most) everything about a finite-dimensional vector space. Adding the structure of commuting operators on top of the scalar action enriches the situation a good deal. In particular, building a module from the analogs of vector spaces (free modules) can be done with a finite amount of data but determining that data can be very complicated. 

If we further impose that our operators must satisfy some collection of relations, then one must generally allow for an infinite amount of data to completely capture a module. When working in positive characteristic, it turns out that one can build any module using just a finite amount information starting from slightly different, but still very basic, building blocks. We will discuss the ideas and techniques behind this result. 

Faculty host: Michael Brown

Brief Bio: Matt is a professor at the University of South Carolina. He studies algebraic geometry, with a specific focus on mirror symmetry. Before starting at South Carolina, Matt obtained his Ph.D. at the University of Washington and was a post doc at the University of Pennsylvania, the University of Wisconsin, and the University of Vienna.