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.

To find the job listing search for "mathematics" and "calculus" positions on the human resources website linked to above.

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.

A general question behind the talk is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80â€™s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this talk, we will focus on a class of noncommutative manifolds obtained by deforming certain Riemannian manifolds along a torus action. I will explain how to formulate some basic notions in Riemannian geometry that are often described in local charts (such as the metric tensor, scalar curvature) using the language of functional analysis so that they will survive in the noncommutative setting. The highlight is that under a noncommutative conformal change of metric, we found not only the conformal change of the scalar curvature in Riemannian geometry but also some exciting new features: the quantum part of the curvature which is hidden in the commutative setting. What is more striking is that the quantum part of the curvature is defined by certain entire functions which play a prominent role in many other areas in mathematics (e.g. in the theory of characteristic classes).

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0, 1, 2, 4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in joint work by Andrew Dolphin and the speaker.

Spectral geometry, among other things, asks the question `can one hear the shape of a drum?' To a mathematical object, say a Riemannian manifold,
one can attach its spectrum and one is interested to know to what extent the object can be recovered from its spectrum. The spectral information can be encoded in terms of zeta functions, heat trace, or wave trace. Isometry invariants like volume and total scalar curvature can be obtained as special values of the spectral zeta function (Weyl's law). I shall give a quick introduction to these ideas and will end by giving the first example of two isospectral manifolds which are not isometric. The example, due to Milnor (using some deep work of Ernst Witt based on the theory of modular forms), exhibits two 16 dimensional flat tori which are isospectral but not isometric. This talk will be accessible to all grad students.