## 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.

## INSTRUCTORS

Farzad Fathizadeh

OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS

Sloan 358

626-395-4355

farzadf@caltech.edu

## 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.

## TEXTBOOKS

- John Roe,
*Elliptic operators, topology and asymptotic methods,*second edition, Longman, 1998. - P. B. Gilkey,
*Invariance theory, the heat equation, and the Atiyah-Singer index theorem,*Studies in Advanced Mathematics, CRC Press, 1995.

## 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 |