This is a graduate class in differential geometry.
This semester the subject is the index theorem for generalized Dirac operators and its proof via heat kernels. The syllabus can be found
here.
Time: T/Th 12:30 PM -- 1:45 PM.
Location: MCS B31
Course schedule and notes
In parentheses are the (approximate) sections of the textbook ``Heat kernels and Dirac operators'' by Berline, Getzler, and Vergne = BGV.
I am updating a terse summary of the main topics covered in lectures
here (last updated: 2/25).
January 23: Introduction to the course. Notes.
January 25: Overview of differential geometry. Notes.
January 30: Differential operators and generalized Laplacians. Notes.
February 1: Classification of generalized Laplacians. Notes.
February 6: Formal adjoints, density bundle. Notes.
February 8: Solution to heat equation on Euclidean space. Notes.
February 15: Schwarz kernels. Notes.
February 20: Existence of heat kernels, part I: iterative step. Notes.
February 22: Existence of heat kernels, pat II: formal heat kernels. Notes.
February 27: Clifford algebras, basic definitions
February 29: Clifford algebras, further properties.
March 5: Clifford modules.
March 7: Clifford bundles, Dirac operators.
March 19: Examples of Dirac operators: de Rham, Dolbeault.
March 21: Examples of Dirac operators: spin Dirac operators.
March 26: Symbols and elliptic differential operators.
March 28: The index density of Dirac operators, I.
April 2: The McKean-Singer formula.
April 4: The A-hat genus and other genera.
April 9: The index density of Dirac operators, II.
April 11: Finishing the local proof of the index theorem for Dirac operators, I.
April 16: Finishing the local proof of the index theorem for Dirac operators, II.
April 18: Instances of the index theorem: Gauss--Bonnet, Riemann--Roch.
April 23: A crash course on the equivariant index theorem.
April 30: The family index theorem and QFT.