Fridays 10:00 - 12:00, April 30 - June 18, 2021
(8 lectures)

Lectures delivered via Microsoft Teams on the Taught Course Centre.
Sign up via gradstud@maths.ox.ac.uk for eligible graduate students (Bath, Bristol, Oxford, Warwick, and Imperial).

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
$$
\mathrm{ind}(D_E)=\int_M \widehat{A}(TM)\mathrm{ch}(E).
$$
This first half of the lectures will be devoted to explaining the components of this formula, developing the required functional analysis for elliptic differential operators, and providing the topological and geometric background. Then we will sketch a proof and finally discuss some of its many applications,
in particular to positive scalar curvature metrics.

Synopsis

Vector bundles, Chern classes, covariant derivatives, Chern-Weil theory; Hirzebruch theory of multiplicative sequences; complex representations of Clifford algebras, spinors; globalization to Riemannian manifolds, spin structures; Dirac operators, other fundamental elliptic operators; pseudo-differential operators and elliptic theory, Fredholm operators; excision proof of the Atiyah-Singer index theorem; applications to positive scalar curvature metrics, generalizations of Rokhlin's theorem.

Prerequisites

Basic knowledge in Differentiable Manifolds, Elementary Topology, Functional Analysis

Course outline

Date

Description

April 30

Introduction and overview

Motivation for index theory by explicit examples (torus, circle, projective space); Historical introduction; Dirac's square root of the Laplacian; proof sketch of Hirzebruch signature theorem; applications (Rokhlin, positive scalar curvature)

Literature

[3, Introduction], [4, Introduction], [5, p.120]

Additional resources

May 7

Vector bundles, characteristic classes, and connections

Bordism; Multiplicative sequences and their classification; Construction of genera; Examples: Todd, A-genus, L-genus, Chern numbers

Literature

[8, Chapter 1], [9], and [12, §17 and §19]

May 21

Clifford algebras and spinors

Quadratic forms; Definition of Clifford algebra by universal property; Basic examples; Identification with matrix algebras; Spinor representations

Literature

[4, Chapter 1],
[11, Chapters 2-3]

May 28

Spin structures and Dirac operators

Principal bundles; Associated vector bundles; Orientations and spin structures; Dirac bundles; Examples of Dirac bundles; Dirac operators and their basic properties

Literature

[10, Chapter II, §1 - §5],
[13, Chapters 3 and 4]

June 4

Elliptic theory and the pseudo-differential calculus

P.B. Gilkey, Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Mathematics Lecture Series 11, Publish or Perish, Inc., Wilmington, DE, 1984.

M. Gromov and H.B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196.