Corina Calinescu (NYC College of Technology, CUNY)
Abstract: A remarkable feature of the theory of vertex operator algebras has been its connection to Lie algebras, combinatorics and modular forms, which led to fundamental results and conjectures in mathematics. Standard (integrable highest weight) modules, and certain subspaces called principal subspaces, for affine Lie algebras have vertex operator constructions. These constructions have been studied in conjunction with combinatorial identities, such as the Rogers-Ramanujan partition identities. In this talk we discuss vertex algebraic structure of principal subspaces of standard modules for affine Lie algebras. This is based on joint works with Lepowsky, Milas, Penn and Sadowski.
Karen Smith (University of Michigan)
Title: Resolution of Singularities
Abstract: Algebraic varieties are geometric objects defined by polynomials---you have known many examples since high school, where you learned that a circle can be defined by a polynomial equation such as $x^2+y^2=1$. Polynomials can define incredibly complicated shapes, such as a mechanical arm in medical software or Woody's arm in toy story, but yet they can be easily manipulated by hand or computer. For this practical reason, algebraic geometry---the study of algebraic varieties and the equations that define them--- is a central research area within modern mathematics. It is also one of the oldest and most beautiful. In this talk I hope to share my love of the subject, which stems from the way the geometry and algebra interact, including some open problems and my favorite tools for attacking them.
Francis Edward Su (Harvey Mudd College)
Abstract: Sperner's lemma is a combinatorial statement equivalent to the Brouwer fixed point theorem, a famous topological theorem. I will trace the history of a generalization of Sperner's lemma to polytopes, that began as a senior thesis project in 2000 and has spawned interesting several applications to cake-cutting, to minimal triangulation of polytopes, and to the game of Hex.