Geometry working seminar

Talk 1. Ehsaan Hossain, Pure Mathematics Department, University of Waterloo

“Extensions of vector bundles”

E is an extension of M by P if there is a short exact sequence 0 → M → E → P → 0. If these are modules then E/M ≃ P; if these are (holomorphic) vector bundles then locally this is a split extension, but of course globally the extension can be more complicated. Goal of the talk: see how these extensions correspond to certain cohomology classes.

Talk 2. Ritvik Ramkumar, Pure Mathematics Department, University of Waterloo

“The Hodge Theorem: Part I”

Let M be a compact complex manifold. The famous Hodge Theorem states that the Dol- beault cohomology groups of M are finite dimensional and gives a very useful L2 orthogonal decomposition for the forms on M. Using the Hodge Theorem one can prove beautiful results such as Serre Duality, the Hodge decomposition for Khler manifolds, the Hard Lefschetz Theorem etc. In a series of talks we will discuss the Hodge Theorem and go over some of these results. In this first talk we will describe various notions from Linear Algebra such as adjointness, extending inner products to the exterior algebra etc. We will also introduce all the operators and the cohomology groups involved in the Hodge Theorem. Time permitting we will start to talk about the Fourier analysis involved in the proof of the Hodge Theorem.