Date & Time: Thursday, January 21, 2016, 15:30-17:30
Venue: Room 216
Title: Chern-Weil theory
Speaker: Saurav Bhaumik, IIT Bombay
Abstract: We will define principal bundles, connections and curvature. With the basics defined, we will construct the Chern-Weil homomorphism. Let E be a principal G-bundle on M with a connection D. Let F be the curvature of D, and g=Lie(G). The Chern-Weil homomorphism associates to each Ad-invariant polynomial on g, a well defined cohomology class in the de Rham cohomology H_{dR}^*(M). Let P be an Ad-invariant homogeneous polynomial of degree k on g. The Chern-Weil image of P is given by the closed 2k-form P(F^{2k}). Its class in H^{2k}_{dR}(M) does not depend on the choice of the connection. This class is functorial. We will conclude with a few examples.