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SF3716 The Atiyah-Singer Index Theorem, graduate course, spring 2022

Meetings: the best possible time seems to be 

  • Wednesdays 15:15-17:00.
    --> No lecture Wednesday May 25 (collision with colloquium and Schock symposium). Final two lectures June 1 and June 8.

We will use this time from Wednesday Feb 9. The meetings will take place

Mail me (dahl@kth.se) if you plan to follow the course, in particular to get access to the course folder.


Course description. The Atiyah-Singer index theorem provides a fundamental connection between differential geometry, partial differential equations, differential topology, operator algebras, and has links to many other fields.

The fundamental observation is that the index of Fredholm operators is a homotopy invariant. Examples of such Fredholm operators are elliptic differential operators on manifolds. The Atiyah-Singer index theorem tells us how the analytic index of an elliptic operator is computed in terms of topological data of the underlying manifold. Since elliptic operators on manifolds often appear related to geometric structures on the manifold, the theorem allows for deep conclusions about the geometry and topology of manifolds. Classical results as The Gauss-Bonnet, Hirzebruch, and Riemann-Roch theorems can be concluded as special cases and will be covered in the course.  

The goal of this graduate level course is to understand the Atiyah-Singer index theorem and its applications. The goal is also to give a background in the different areas of mathematics involved, such as functional analysis; elliptic differential operators on manifolds; Clifford algebras, spinors, Dirac operators; vector bundles and characteristic classes.

This is a ``broad graduate course'' which is supposed to be accessible and meaningful for all graduate students in mathematics.

Syllabus.

  • Index theory in functional analysis.
  • Elliptic operators on manifolds.
  • Dirac operators.
  • Vector bundles, characteristic classes, K-theory.
  • The Gauss-Bonnet, Hirzebruch, Riemann-Roch theorems.
  • The Atiyah-Singer index theorem.
  • First proofs.
  • The heat kernel proof of the index theorem in detail.
  • Applications to obstructions to positive scalar curvature, dimensions of moduli spaces, Donaldson and Seiberg-Witten invariants.

Prerequisities. Courses on functional analysis, differential geometry, algebraic topology are recommended as prerequisities, but it is not strictly necessary to have read all these courses.Prerequisities.

Literature. Parts of different books will be used, primarily

  • Berline, Getzler, Vergne: Heat kernels and Dirac operators
  • Gilkey: Invariance Theory, the heat equation and the Atiyah-Singer index theorem
  • Hörmander: Riemannian Geometry lecture notes
  • Roe: Elliptic operators, topology and asymptotic methods

All literature will be provided.

Examination. The examination is in the form of homework problems and presentations.


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