Geomatikum 428. 14:30-16:00PM every Friday.
1. Apr 11: Motivation of equivariant cohomology and definition of G^∗-modules. [GSB] Chapter 1, 2.1-2.2
2. Apr 25: Equivariant cohomology of G^∗-algebras. [GSB] Chapter 2.3-2.5
3. May 2: Weil model and Cartan model. [GSB] Chapter 3, Chapter 4.1-4.5
4. May 9: Cartan's formula and Spectral sequences. [GSB] Chapter 4.6-4.7, Chapter 5, Chapter 6.
5. May 16: Mathai-Quillen formalism and fermionic intergration. [GSB] Chapter 7
6. May 23: Equivariant characteristic classes and splitting principle. [GSB] Chapter 8
7. Jun 6: Symplectic reduction and equivariant Duistermaat-Heckman theorem. [GSB] Chapter 9
8. Jun 13: Geometric quantization and Borel-Weil-Bott theorem. [B] II.1-2, [J] Chapter 11.
9. Jun 20: Thom isomorphism and proof of (abelian) Atiyah-Bott localization. [GSB] Chapter 10
10. Jun 27: Introduction to (equivariant) K-theory. [L] Chapter 1,4, [Tal] Talk 2
11. Jul 4: Clifford algebras, Spin groups and their representations. [LM] Chapter I.1-I.5
12. Jul 11: Atiyah-Bott-Shapiro construction and spin structure . [LM] Chapter I.9, II.1,II.2.
13. Jul 18: Spinor bundle, connections and Dirac operator [LM] Chapter II.3-II.5.
Extra talk: Atiyah-Singer index theorem [L] Chapter 2 ,3 or [Tal] Talk 3 or [Fr] Chapter 4 or [LM] III.1, III.13
Homework: https://www.youtube.com/watch?v=AJHKp9kYm90 (Don't read comments)
Main reference: Victor W. Guillemin, Shlomo Sternberg, Jochen Brüning, Supersymmetry and Equivariant de Rham Theory [GSB]: Good introduction to (Borel) equivariant cohomology, also contains some applications in physics/symplectic geometry.
Other helpful references:
Loring W. Tu , Introductory Lectures on Equivariant Cohomology [Tu]: A more ground-up introduction to Borel equivariant cohomology and contains a detailed proof of equivariant de Rham theorem.
David Anderson and William Fulton, Equivariant Cohomology in Algebraic Geometry: As per title. Focused on localization techniques used in enumerative geometry.
Main reference: Lawson, Michelsohn, Spin Geometry: Comprehensive treatment of index theorem from K-theory. [LM]
Other helpful references:
Bernhelm Booss and David Bleecker, Index Theory with Applications to Mathematics and Physics: Very gentle and comprehensive introduction to index theorem and its applications.
Gregory D. Landweber, K-Theory and Elliptic Operators [L]: Concise introduction to the celebrated Atiyah-Singer index theorem from K-theoretic point of view.
Berline, Getzler, and M. Vergne, Heat Kernels and Dirac Operators: Advanced treatment of index theorem from analytic point of view.
Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization. [B]
Dan Freed, The Atiyah-Singer index theorem [Fr]
Inna Zakharevich , K-theory and Characteristic Classes: A homotopical perspective
Arun Debray, M392C: K-Theory [D]
Max Karoubi, Lectures on K-Theory [K]
Liviu I. Nicolaescu, Notes on the Atiyah-Singer Index Theorem [N]
Araminta Amabel, Arun Debray, Peter J. Haine, Differential Cohomology: Categories, Characteristic Classes, and Connections. [ADH]
Basic notions in algebraic topology: Universal/principal bundles, cohomology...
Basic notions in differential geometry: vector bundles, connection/curvature forms, Lie groups/algebras...
A few notions in symplectic geometry: Hamiltonian group actions, moment maps, coadjoint orbits...
[Tu], Chapter 2,3,5,8.
[Tu], Chapter 10-17.
Lisa Jefferey et al. Hamiltonian Group Actions and Equivariant Cohomology, [J] Chapter 1-5