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.

In the present talk, I will review, from a mathematical viewpoint, the computations of instanton partition functions of supersymmetric gauge theories in four dimension by means of quiver varieties and their relations to representation theory of vertex algebras.

By extending the canonical trace of Kontsevich-Vishik to Connes' pseudodifferential operators on noncommutative tori, we study various spectral invariants associated to elliptic operators in this setting. We also consider a family of Cauchy-Riemann operators over noncommutative 2-torus and using the machinery of canonical trace, we compute the curvature form of the associated Quillen determinant line bundle.

We will present a theorem due to R.M.F. Moss which says, roughly, that 3-fold Massey products of permanent cycles in the Adams spectral sequence converge to the corresponding 3-fold Toda brackets in stable homotopy.

The Kepler conjecture asserts that no packing of congruent
balls in space can have density greater than the familiar cannonball
arrangement. If every logical inference of proof has been checked all
the way to the fundamental axioms of mathematics, then we say that the
proof has been formally verified. The Kepler conjecture has now been
formally verified by computer, in a massive cloud computation. This
talk will report on this and other massive formal verification projects.

The index of a bounded operator $T\in B(H)$ of a Hilbert space $H$ is defined as the difference between the dimensions of kernel and cokernel. That is,
$${\rm Ind}(T):=\dim(\ker T)-\dim({\rm coker}T)$$
This index, if defined, is called the Fredholm index.
The Fredholm index of an operator on a finite dimensional Hilbert space $H$ by the dimension theorem in linear algebra. However, the case of infinite dimensional Hilbert spaces requires more delicate analysis and an operator with nonzero index exists.
The celebrated local index formula in noncommutative geometry (Connes and Moscovici 1995) relates the index of Dirac type operators and the residue cocycle in the cyclic cohomology. In the classical case, this formula equates topology and geometry. In my talk, I will prove two special cases of local index formula following closely the chapter 5 in Noncommutative geometry and particle physics by Walter Van Suijlekom. If the time is allotted, I will demonstrate the strength of the formula using simple classical spectral triples such as the circle $S^1$.