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 AtiyahSinger 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 pseudodifferential calculus 
June 11 
AtiyahSinger index theorem 
June 18 
Applications to positive scalar curvature 

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.

