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.

Upcoming Events

Each week the Department of Mathematics hosts a wide variety of seminars and events. For a comprehensive list of events, please consult our departmental calendar.

Starting from the works of Nakajima and Grojnowski, moduli spaces and stacks of sheaves on surfaces represent wonderful tools for the study of vertex and quantum algebras and their representations from a geometric point of view. For example, Schiffmann and Vasserot proved that the equivariant K-theory of the stack of zero-dimensional sheaves on $\mathbb C^2$ has an associative algebra structure and is isomorphic to the positive part of quantum toroidal algebra of type $\mathfrak{gl}(1)$; moreover, it acts on the equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. Their result can be seen as a K-theoretic version of Nakajima-Grojnowski cohomological result for Hilbert schemes of points.
In the present talk, I would like to describe a new conjectural approach to the study of quantum toroidal algebras of type $\mathfrak{gl}(k)$ based on the study of algebra structures on the K-theory of the stacks of torsion sheaves over other noncompact surfaces (e.g. the stack of sheaves on the minimal resolution of the Du-Val singularity $\mathbb{C}^2/\mathbb{Z}_k$â€‹, supported at an exceptional curve)â€‹. (This is a work in progress with Olivier Schiffmann.)

Let $E$ be a closed subset in the complex plane with connected complement. We define $A(E)$ to be the class of all complex continuous functions on $E$ that are holomorphic in the interior $E^0$ of $E$. The remarkable theorem of Mergelyan shows that every $f\in A(E)$ is uniformly approximable by polynomials on $E$, but is it possible to realize such an approximation by polynomials that are zero-free on $E$? This question was first proposed by J.Anderson and P.Gauthier. Recently Arthur Danielyan described a class of functions for which zero-free approximation is possible on an arbitrary $E$. I am intending to talk about the generalization of his work on Riemann surfaces.