Registration for Spring term opens Thursday, February 23, 2016.
This page was last updated: .
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.