# Index Theory of Elliptic Operators

## 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.