TCC Course TT2021
Index Theory of Elliptic Operators
Sign up via gradstud@maths.ox.ac.uk for eligible graduate students (Bath, Bristol, Oxford, Warwick, and Imperial).
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Synopsis
The Atiyah-Singer index theorem is one of the cornerstones of modern geometry. It relates the number of solutions
of elliptic linear differential equations to the topology of the underlying manifold. For a Dirac operator
on a spin manifold $M$ acting on $E$-valued spinors the index formula states that
$$
\mathrm{ind}(D_E)=\int_M \widehat{A}(TM)\mathrm{ch}(E).
$$
The first goal of the course willl be to explain all of the components of this formula, and to develop the
functional analysis and geometry background. Then we will sketch a proof and discuss its many applications,
in particular to positive scalar curvature metrics.
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Prerequisites
-
Differentiable manifolds
- Elementary topology
(Basic knowledge of both is sufficient)
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Course outline
10:00 - 12:00 on Fridays on MS Teams via TCC
| Date |
Description |
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| April 30 |
Introduction and overview |
| May 7 |
Vector bundles, characteristic classes, and connections |
| May 14 |
Genera and multiplicative sequences |
| May 21 |
Clifford algebras and spin structures |
| May 28 |
Spinors and Dirac operators |
| June 4 |
Elliptic theory and the pseudo-differential calculus |
| June 11 |
Atiyah-Singer index theorem |
| June 18 |
Applications to positive scalar curvature |
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Literature
- B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.
- J. Roe, Elliptic operators, topology and asymptotic methods (Second edition), Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998.
- M. Gromov and B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196.
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