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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).

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.

Prerequisites

  • Differentiable manifolds
  • Elementary topology
(Basic knowledge of both is sufficient)

Course outline

 10:00 - 12:00 on Fridays on MS Teams via TCC

Date Description

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

Literature

  1. B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.
  2. J. Roe, Elliptic operators, topology and asymptotic methods (Second edition), Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998.
  3. 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.