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| Advanced Instructional School on Atiyah-Singer Index TheoremFunded byNational Board for Higher Mathematics Venue : Department of Mathematics, IIT Bombay 6th July - 1st August, 2009 |
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| Advanced Instructional School on Atiyah-Singer Index TheoremFunded byNational Board for Higher Mathematics Venue : Department of Mathematics, IIT Bombay 6th July - 1st August, 2009 |
| Ravi Kulkarni | IIT, Bombay |
| Nitin Nitsure | TIFR, Mumbai |
| M.S. Raghunathan | TIFR, Mumbai |
| A. Ranjan | IIT, Bombay |
| Harish Seshadri | IISc, Bangalore |
| A. R. Shastri | IIT, Bombay |
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Basic Differential Geometry. Manifolds,
differential forms, Stokes theorem, tangent and cotangent bundles,
Riemannian metrics, connections on vector bundles, jet bundles
of vector bundles, linear differential operators between vector bundles,
Peetre theorem, Laplacian, statement of Hodge theorem.
Overview of singular cohomology, CW complexes, vector bundles, characteristic classes (Chern and Euler) via axioms, existence of characteristic classes via classifying spaces, elementary facts about cobordism and Morse theory, Thom's theorem about spherical cohomology classes on manifolds. Operations with vector bundles, pullbacks, homotopy invariance, clutching construction, collapsing a closed subspace, the relative K functors, long exact sequence for a pair, Bott periodicity, half-exact functors, isomorphism of rational K theory with cohomology with rational coefficients via Chern character, Thom isomorphism in (integral) K theory for complex vector bundles. Twisted signature operator, Thom isomorphism for K tensored with rationals, comparison theorem for Thom isomorphism in rational K theory and rational cohomology, Dolbeault complex, Hodge Index Theorem, Hirzebruch-Riemann-Roch Theorem, Dirac operators. Elliptic operators, a priori inequalities, finiteness theorem for kernel and cokernel, the heat equation, the heat kernel and its asymptotics. Proof of the Atiyah-Singer Index Theorem |
