Registration for Spring term opens Thursday, February 23, 2016.
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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.
Acquaintance with manifolds, exterior derivative, Riemannian metrics, and the spectral theorem for compact self-adjoint operators on a Hilbert space, will be very useful.
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.
There is no homework for this course.
- 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.
|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|
|Thur Apr 27th|
|Tues May 2nd|
|Thur May 4th|
|Tues May 9th|
|Thur May 11th|
|Tues May 16th|
|Thur May 18th|
|Tues May 23rd|
|Thur May 25th|
|Tues May 30th|
|Thur June 1st|