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.
The generalized Wielandt inequality relates the condition number of a complex matrix and the amount by which the action of the matrix may distort angles. In Banach spaces, it leads to a notion of norm equivalence that is stronger than the usual, topological, equivalence of norms.
Speaker's web page: http://www-home.math.uwo.ca/~sinnamon/
Ricci curvature is a shadow of the Riemann curvature tensor. It appears in one of the most important equations of physics, that is in Einstein's field equations. Motivated by Gilkey's local spectral invariants in Riemannian geometry, one can show that the Ricci operator can be expressed in terms of residues of the spectral zeta function for the de Rham complex. We use this observation to propose a definition of the Ricci curvature for noncommutative manifolds. I will indicate how this Ricci curvature can be computed for curved noncommutative 2-torus. Based on joint work with R. Floricel, and A. Ghorbanpour (arXiv: 1612.06688).
Hessenberg varieties are subvarieties of the full flag variety. In
this talk, I will concentrate on Lie type A. I will talk about a flat
degeneration of a regular semisimple Hessenberg variety to a regular
nilpotent Hessenberg variety, and I will explain how we can use this flat
family to compute some Newton-Okounkov bodies of the Peterson variety of
dimension 2. Along the way, we will also see that any regular nilpotent
Hessenberg variety is a local complete intersection; this is a
generalization of a result in Erik Inskoâ€™s PhD thesis. This is a joint work
with Lauren DeDieu, Federico Galetto, and Megumi Harada.
Let $M$ be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. Lempert-Szoke and Guillemin-Stenzel have given canonical methods to construct adapted complex structures in neighborhoods of the zero section, equipped with a solution of the Homegeous Complex Monge-Ampere equation (HCMA) related to the geometry of $M$. These complex manifolds are called Grauert tubes. This structure is called entire if the structure may be extended to the whole of $TM$. We describe a circle of problems related to determining whether an entire Grauert tube is an affine algebraic manifold with ring of polynomials intrinsically distinguished by the HCMA exhaustion. We also discuss the relationship of this construction to the Paley-Wiener type theorem of Boutet de Monvel, and the relationship to eigenfunctions of the Laplace operator on $M$. Finally, we discuss the smallest dimensional cases, namely the two sphere, and invariant metrics on the three sphere, thought of as $SU(2)$.
Speaker's web page: https://lsa.umich.edu/math/people/faculty/dburns.html