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.
An action of a group $G$ on a space $X$ is said to be equivariantly formal if
the induced map from Borel equivariant cohomology of the action to singular
cohomology of $X$ is surjective. This situation is much to desired in various
geometric settings, where it can allow the integral of an invariant
function to be reduced to one over a lower-dimensional or even finite
It turns out (work of Fok) one can approach the question in terms of
equivariant $K$-theory: the action of a compact Lie group $G$ on a finite $G-CW$
complex $X$ is equivariantly formal with rational coefficients if and only if
some each power of every vector bundle over $X$ admits a stable equivariant
structure. This correspondence allows us an alternate proof, with a
slightly stronger conclusion, of a result of Ademâ€“GÃ³mez on the equivariant
$K$-theory of actions with maximal-rank isotropy.
In the case of the left translation ("isotropy") action of a connected
group $H$ on a homogeneous space $G/H$, the correspondence also allows us to
more simply reobtain results of Goertschesâ€“Noshari on generalized symmetric
spaces. In the realm of rational homotopy theory, we are able to show that
equivariant formality of the isotropy action implies $G/H$ is a formal space,
allowing us to improve a sufficient condition of Shigaâ€“Takahashi to an
This work is joint with Chi-Kwong Fok.
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.