Abstract: 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.