Index Theory of Elliptic Operators

Markus Upmeier

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).

Lecture notes
Last update: May 29, 2021

Overview and synopsis
Course outline and additional resources
Bibliography

Overview

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]

May 7 Vector bundles, characteristic classes, and connections
Definition of vector bundle; Transition functions; Connections; Curvature; Invariant polynomials; Chern-Weil homomorphism; Chern classes

Literature

[1, Chapter 1], [12, §2 and Appendix C]

May 14 Genera and multiplicative sequences
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
June 11 Atiyah-Singer index theorem
June 18 Applications to positive scalar curvature

Bibliography

1
M.F. Atiyah, K-theory, W.A. Benjamin, New York--Amsterdam, 1967.

2
M.F. Atiyah and I.M. Singer, The index of elliptic operators. I, Ann. of Math. 87 (1968), 484–530.

3
D.D. Bleecker and B. Booß-Bavnbek, Index theory—with applications to mathematics and physics, International Press, Somerville, MA 2013.

4
T. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics 25, AMS, Providence, RI, 2000.

5
I.M. Gel'fand On elliptic equations Uspehi Mat. Nauk 15 (1960), 113–123.

6
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.

7
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.

8
F. Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics 131, Springer-Verlag, Berlin, 1995.

9
F. Hirzebruch, T. Berger, and R. Jung, Manifolds and modular forms, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992.

10
H.B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton, NJ, 1989.

11
E. Meinrenken, Clifford algebras and Lie theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Springer, Heidelberg, 2013.

12
J.W. Milnor and J.D. Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press, Princeton, N. J., 1974.

13
J. Roe, Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Mathematics Series 395, Second Edition, Longman, Harlow, 1998.