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.

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 $S^1$.

A flat output of a control system allows to express its state and its inputs as a function of the flat output and its derivatives. It can be used, for example, to solve motion planning problems. We propose a variation of the definition of flatness for linear differential systems to linear differential-delay systems with time-varying coefficients which utilises a prediction operator $\pi$. We characterize $\pi$-flat outputs and provide an algorithm to efficiently compute such outputs.
(Joint work with Jean Levine, Felix Antritter and Franck Cazaurang)

T-spectra, or spectrum objects with generalized "suspension" parameters, first appeared in the construction of motivic stable categories. The motivic stable model structure lives within a localized model structure of simplicial presheaves in which the affine line is formally collapsed to a point, and the suspension object is the projective line. Because of these constraints and limitations of the tools then at hand, the original construction of the motivic stable category was technical, and made heavy use of the Nisnevich descent theorem.
This talk will begin with a general introduction to the concepts around T-spectra. I shall display a short list of axioms on the parameter object T and the ambient f-local model category which together lead to the construction of a well behaved f-local stable model structure of T-spectra. Examples include the motivic stable category, but the construction is much more general. The resulting stable category has many of the basic calculational features of the motivic stable category, including slice filtrations. The overall construction method is to suitably localize an easily defined strict model structure for T-spectra. The localization trick is an old idea of Jeff Smith, but assumptions (the axioms) are required for the recovery of normal features of stable homotopy theory from the localized structure.