Ma 191c-sec3:  The Dirac Operator and the Atiyah-Singer Index Theorem

ANNOUNCEMENTS

Registration for Spring term opens Thursday, February 23, 2016.

This page was last updated:  .


COURSE DESCRIPTION

This course presents the heat equation proof of the Atiyah-Singer index theorem. The following topics will be covered: spin geometry, Dirac operator, pseudodifferential calculus, Fredholm operators, the heat kernel method for computing the Seeley-de Witt coefficients and the index of an elliptic operator on a closed manifold, the McKean-Singer index formula, the Hodge decomposition theorem, characteristic classes, the harmonic oscillator, Mehler's formula, and the Atiyah-Singer index theorem.


PREREQUISITES

Acquaintance with manifolds, exterior derivative, Riemannian metrics, and the spectral theorem for compact self-adjoint operators on a Hilbert space, will be very useful.


SCHEDULE

Tuesday and Thursday, 2:30 - 3:55 p.m., 159 Sloan.


INSTRUCTORS

Farzad Fathizadeh
OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS
Sloan 358
626-395-4355
farzadf@caltech.edu


TA's

There is no TA for this course.


OFFICE HOURS

Friday 4 to 5 p.m., Sloan 358.


POLICIES

Grades

Based on final presentation and class attendance. Suggested references for choosing the topic of the final presentation will be posted at the bottom of this page.

Homework Policy

There is no homework for this course.


TOPICS COVERED

Please refer to the Course Description.


TEXTBOOKS


LECTURE NOTES

Date Description
Tues Apr 4th An overview of the material to be covered in the course, and discussion of the main ideas and techniques that will be used for the proof of the index theorem
Thur Apr 6th Connections on vector bundles, Levi-Civita connection, Riemann curvature tensor and its properties
Tues Apr 11th Clifford algebras, Clifford bundles over Riemannian manifolds, the Dirac operator of a Clifford bundle
Thur Apr 13th Hodge star operator, an example of a Clifford bundle and its Dirac operator: the exterior bundle of the cotangent bundle and the Hodge-de Rham opertor
Tues Apr 18th Representation theory of finite groups, the spin representation as the unique irreducible representation of the Clifford algebra
Thur Apr 20th Complex manifolds, explicit realization of the spin representation, spin\(^c \) manifolds
Tues Apr 25th Clifford algebra as a super-algebra and its super-center, Pin and Spin groups, double covering of the special orthogonal group
Thur Apr 27th Idenitification of the Lie algebra of the Spin group with a linear subspace in the Clifford algebra and its canonical isomorphism with the Lie algebra of the special orthogonal group, spin structures on manifolds
Tues May 2nd Local description of connections using matrices of 1-forms, an explicit local formula for the Dirac operator of a spin manifold, pseudodifferential symbol of the Dirac operator
Thur May 4th Fourier transform, symbol of Differential and pseudodifferential operators, Sobolev and Rellich lemma, elliptic operators
Tues May 9th Derivation of the small-time asymptotic expansion for the trace of the heat kernel of a positive elliptic differential operator on a vector bundle on a compact manifold
Thur May 11th The McKane-Singer index theorem, spectral decompositon for positive elliptic differential operators, the heat equation, the heat kernel and fundamental solutions to the heat equation
Tues May 16th Uniqueness of the fundamental solution of the heat equation, harmonic oscillator and its spectral decomposition, Mehler's formula
Thur May 18th Chern-Weil theory and construction of characteristic classes
Tues May 23rd Independence of the cohomology class of characteristic classes from the choice of the connection, Chern classes, the Chern character and the A-hat class, the Weitzenboch formula, and the Bochner theorem
Thur May 25th The square of the Dirac operator of a spin manifold, the index problem, and the super trace of the Clifford action on the spin representation
Tues May 30th Proof of the Atiyah-Singer index theorem
Thur June 1st Proof of the index theorem
Student presentations: June 1st from 4 to 5:30 and June 2nd from 3 to 6 pm

HOMEWORK

Due Date Homework Solutions

EXAMS

 


PAPERS FOR PRESENTATIONS