Tim Hosgood: Simplicial Chern–Weil Theory
Time: Wed 2020-12-02 15.30 - 16.15
Location: Zoom, meeting ID: 657 9019 8929
Participating: Tim Hosgood
Fibre bundles exist in many settings, and can often be better understood by studying their characteristic classes. These cohomology classes generally satisfy similar axioms, and can usually be calculated in a number of ways. One technique from differential geometry is that of Chern–Weil theory, where we consider invariant polynomials evaluated at the curvature of a connection. This technique gives very concrete representatives of the classes, but fails to generalise to the setting of coherent sheaves on complex-analytic manifolds for two reasons: (1) global holomorphic connections rarely exist; and (2) coherent analytic sheaves are rarely resolved by locally free sheaves. Both of these problems were solved by Green (1980) via simplicial methods. In this talk, we describe these methods as specific examples of a more formal framework, and allude to how we can obtain an equivalence of (∞,1)-categories that helps us to better understand coherent analytic sheaves.
Zoom Notes: The meeting ID is 657 9019 8929 and the passcode is 3517257.