The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
Polynomial identity algebras (or PI-algebras) can be seen as generalizations of both commutative and finite dimensional algebras which maintain many ``nice'' properties. In this talk, we give a short introduction to Lie PI-algebras, focusing on the study of multilinear identities. Using these techniques, we demonstrate the existence of an identity for Lie algebras having certain actions on them; such actions arise naturally, for example, when the Lie algebra is graded by a finite group.
I will provide a brief introduction to impartial game theory consisting of examples of games as well as elementary operations on them. This will be followed by a presentation of their syntactic counterparts in Coq, culminating in formal proofs of several basic theorems.
An amazing discovery of physicists is that there are many seemingly
quite different quantum field theories that lead to the same
observable predictions. Such theories are said to be related by
dualities. A duality leads to interesting mathematical consequences;
for example, certain K-theory groups on the two spacetime manifolds
have to be isomorphic. We will explain how some of these K-theory
isomorphisms predicted by physics correspond to certain cases of
the Baum-Connes Conjecture, or to equivalences of
derived categories of twisted coherent sheaves.