The Graduate Student Seminar is put on by the Mathematics Graduate Student Association . GSS meets approximately every other Monday.
For the Fall 2014 semester, the GSS meets at 4:00p in 125 HH.
To volunteer to give a talk, or for anything else regarding the seminar, contact Dominic Culver.
|Monday, September 8||Katrina Barron||An Introduction to the Algebraic Aspects of Conformal Field Theory and Applications|
|Monday, September 22||Augusto Stoffel||2-Dimensional Topological Field Theories|
|Monday, Octobr 6||Edward Burkard||A Primer on Pseudoholomorpic Curves|
Conformal Field Theory (CFT) is an attempt to unify all fundamental forces, including gravity, by modeling particles as vibrating strings. In the genus-zero, two-dimensional setting, vertex operator algebras (VOAs) describe the particle interactions. Independently from physics, VOAs were discovered in mathematics in the study of representations of infinite-dimensional Lie algebras and the Monster finite simple group. This study led to the surprising connection between CFT, VOAs and number theory. I will briefly give a flavor of some of the mathematics involved.
We will show that the notion of a topological field theory, in one of its simplest variations (namely, 2-dimensional and oriented), recovers a classical notion from algebra, that of a Frobenius algebra. This fact is a well-known “folk theorem” but, as we will see, it has a nontrivial proof relying on Morse theory.
I will begin with an overview of almost complex manifolds and complex manifolds. Next, I will define pseudoholomorphic curves (or (J-)holomorpic curves) and I will try to convince you that these are just the usual notion of a holomorphic function in a special case. After that, I will give some basic definitions in symplectic geometry and relations with almost complex structures. Finally, I will state some results in symplectic geometry which are applications of holomorphic curves, one of which can be thought of as a generalization of the Riemann Mapping Theorem.