Regarding the space of Hermitian matrices as an N^2 dimensional real vector space with nondegenerate bilinear form given by the trace, we may apply Feynman's theorem to compute matrix integrals. First I will show how to evaluate such integrals by a sum over compact oriented surfaces with boundary, and then I will use this expansion to prove a theorem of t'Hooft which states that in the limit for large N of such integrals the sum only depends on the contribution of planar connected fat graphs.

After defining equivariant (co)homology for torus actions, I will present an equivariant version of Poincaré-Alexander-Lefschetz duality and relate it to an old result of Duflot.

Then I will turn to syzygies in equivariant cohomology. Syzygies are modules (over a polynomial ring) that interpolate between torsion-free and free modules. I will recall how syzygies are related to equivariant homology, the Atiyah-Bredon sequence and the equivariant Poincaré pairing. For actions on manifolds I will then give a "geometric criterion" that characterizes such syzygies in terms of the orbit space together with its stratification by orbit dimension. At the end I will discuss the existence of "maximal" syzygies for compact orientable manifolds. Here an interesting connection with singularities of real algebraic varieties appears.

This is joint work with Chris Allday and Volker Puppe.

Abstract: The fundamental Kohn's Decomposition Theorem relates cohomology groups of forms on compact subdomains of complex manifolds (e.g. pseudoconvex), to finite-dimensional spaces of harmonic forms on these subdomains. In my talk I will introduce a variant of Kohn's theorem for forms defined on non-compact subdomains, and satisfying additional constraints on their growth along discrete subsets (joint work with Alex Brudnyi). Its proof is based on a quite useful technique for dealing with infinite-dimensional holomorphic Banach vector bundles, which I will also describe. Finally, I will demonstrate how infinite-dimensionality of vector bundle, combined with Oka principle, can lead to better results than in the finite-dimensional case.

To celebrate the year's end, we shall have two talks this Tuesday. The first talk is in our Pizza Seminar series and will be given by Mitsuru Wilson and the second will be the last lecture in our Discovery Cafe weekly meetings by Masoud Khalkhali. We shall then all go to the grad club for Pizza, courtesy of Math Department! Tea will be served in the lounge between the two talks.

The first talk is a gentle and friendly introduction to evolution of geometric thought through history of mathematics, culminating in some current ideas on noncommutative spaces.

The second talk is an introduction to Shor's fast factorization quantum computing algorithm and some of its physics and mathematics background. Please check the one and only Pizza Seminar blog http://pizzaseminaruwo.blogspot.ca/ for more details!

In a seminal paper of 1985 Gromov proved that any compact Lagrangian submanifold of $C^n$ admits a nonconstant analytic disc attached to it. I will outline Alexander's proof of this result and discuss possible generalizations for immersed Lagrangian manifolds.

This is the Euclidean Wick rotated analogue of Feynman's formula for the propagator. Unlike the latter, it can be rigorously proved. I shall first define the Wiener measure on the space of continuous paths and then prove the formula. I end with a few examples and applications.

This talk is based on ongoing work with X. Tang and H. Tseng that grew out of my attempt to understand a certain duality for gerbes on orbifolds that Tang and Tseng studied in their paper "Duality theorems of etale gerbes on orbifolds". Our new approach is more conceptual, allowing the definition of duality to be extended in greater generality. In the talk I will explain gerbes from the point of view of bitorsors and sketch the constructions involved in duality, both original and twisted.