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.

We are covering the topics: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula,
---Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula.

Milnor-Witt K-theory arises in the Morel-Voevodsky homotopy theory over a field and plays a role in the classification of vector bundles over smooth schemes. Morel in collaboration with Hopkins discovered a nice presentation of these groups, which has been recently generalized by Changlong Zhong, Stephen Scully and myself to semilocal rings which contain an infinite field. In my talk I will discuss this result and also present some applications of these groups.

"Down, down, down. Would the fall never come to an end! I wonder how many miles I have fallen by this time? she said aloud. I must be getting somewhere near the center of the earth", wrote Charles Dodgson, the English writer and mathematician, of Alice's fantastic adventure into a bizarre world of underground creatures. I shall retell the story of our centennial adventure into the relativistic wonderland, a tall tale of non-sensical creatures that includes blunders, black holes, Schroedinger's cats, and a very vibrant vacuum!

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.