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.
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.
Algebraic groups are a well studied object that arise when one has an algebraic variety with a group structure compatible with the variety. In the same vein algebraic monoids are varieties with a monoid structure imposed on them. An interesting thing happens to certain algebraic groups and algebraic monoids called the Bruhat deomposition, which provides a wealth of knowledge about the groups/monoids in terms of double cosets. We'll take a look at two collections of algebraic monoids and their Bruhat decompositions, and determine generating functions for the "sizes" of their Bruhat decompositions.
In this talk I will define spin structure and spin manifolds and give some examples. I will also quickly review some notions of Riemannian geometry like connections, curvature; in particular, the Levi-Civita connection. Then I will define the spin connection and using that I will introduce the Dirac operator.
We will outline a certain program for Nakajima quiver varieties, in the cyclic quiver example. The picture includes two algebras that act on the K-theory of these varieties: one is the original picture by Nakajima, rephrased in terms of shuffle algebras, and the other one is the Maulik-Okounkov quantum toroidal algebra. The connection between the two is provided by the action of certain operators in the so-called "stable basis", and we will present formulas for this action. These formulas can be perceived as a generalization of Lascoux-Leclerc-Thibon ribbon tableau Pieri rules.
There is a result from the 1980's that allows us to describe the equivariant $K$-theory of curves: if $X$ and $Y$ are curves, G is a finite reducible group and $Y = X/G$, then we can write $K_G(X)$ in terms of $K(Y)$ and some representation rings associated to the group. Prof. Dhillon and I have generalized this result to any dimension using the description of the category of coherent sheaves on a root stack given by Borne and Vistolli.