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||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|