Sep 04, 11:0 - 12:30
The NCG learning seminar's topic this year will be Geometric Analysis. One of our goals is
to go through a heat equation proof of the celebrated Atiyah-Singer index theorem. The following topics will be covered:
1. Operators of Dirac type and its main examples,
2. Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula,
3. Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula,
4. The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory,
5. Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch.
6. Approach via path integrals and quantum mechanics,
7. Atiyah-Bott-Lefschetz fixed point formula.
Sep 10, 14:30 - 15:30
In this talk we will describe basic topological properties of the space
of commuting unitary matrices. In particular we will show that they can
be assembled to form a space which classifies commutativity for vector
bundles and which has very interesting homotopy-theoretic properties.