The Atiyah-Singer Index Theorem student seminar met (semi-)regularly on Tuesdays and Thursdays at 3pm, in RLM 12.161 during the summer break of 2013. (Check the schedule below for exact dates.) The seminar was organised by Richard Hughes (contact: rhughes [at] math [dot] utexas [dot] edu) with support and oversight from Andrew Neitzke and Dan Freed.
This aim of this seminar is to investigate the Atiyah-Singer index theorem, which provides a link between functional analysis and differential topology by expressing the 'analytic index' of an elliptic differential operator in terms of data that is topological in nature (the 'topological index'). We will be following John Roe's book Elliptic operators, topology and asymptotic methods, in which he presents a proof of the index theorem based on the asymptotic expansion of the heat kernel.
I have written up some of the review differential geometry presented in Chapter One of Roe's book, with some parts expanded on and other parts condensed. The document is still incomplete, and possibly incorrect in some places -- corrections or clarifications are welcomed. As well as Roe's book, I made use of some lectures on differential geometry given by Craig Hodgson at the University of Melbourne. Any mistakes are all mine, and not the fault of either reference.
The scratch work referenced in the differential geometry review above is here.
The schedule of talks is as follows:
| Date | Topic | Speaker |
|---|---|---|
| 11th June | Characteristic classes (ref: Roe, Ch.2) | Lee Cohn |
| 13th June | Clifford algebras and Dirac operators (ref: Roe, Ch.3) | Tom Mainiero |
| 18th June | Spin Groups I (ref: Roe, Ch.4) | Aaron Fenyes |
| 20th June | Spin Groups II (ref: Roe, Ch.4) | Richard Hughes |
| 25th June | Analytic Properties of Dirac Operators (ref: Roe, Ch.5) | Andrew Lee |
| 27th June | Hodge Theory (ref: Roe, Ch.6) | Rustam Antia-Riedel & Javier Morales |
| 1nd July | Problem Session and Discussion [Meet in 12th floor lounge at 3pm.] | N/A |
| 2nd July | Problem Session and Discussion | N/A |
| 4th July | Day off due to Higgs Boson anniversary. | N/A |
| 9th July | Spin Groups III (ref: Roe, Ch.4) | Rahul Shah |
| 11th July | The Heat and Wave Equations (ref: Roe, Ch.7) | Valentin Zakharevich |
| 16th July | Traces and Eigenvalue Asymptotics (ref: Roe, Ch.8) | Cagri Karakurt |
| 18th July | The Harmonic Oscillator and Non-Compact Manifolds (ref: Roe, Ch.9) | Richard Hughes |
| 23rd July | The Lefschetz Formula (ref: Roe, Ch.10) (Extra notes) | Laura Starkson |
| 25th July | The Index Problem (ref: Roe, Ch.11) | Valentin Zakharevich |
| 30th July | Problem Session and Discussion | N/A |
| 1st August | Problem Session and Discussion | N/A |
| 6th August | The Getzler Calculus (ref: Roe, Ch.12) | Richard Hughes |
| 8th August | The Index Theorem (ref: Roe, Ch.12) | Valentin Zakharevich |
| 13th August | No session. | N/A |
| 15th August | The Hirzebruch-Riemann-Roch Theorem [Alternate room: 11.176] | Lee Cohn |