In the spring semester of 2025 Hessel Posthuma, Mikhail Isachenkov and I are organizing a reading seminar on the local index theorem. This is an informal seminar where the audience learns together the material that is needed for the local index theorem. Among other things, we treat von Neumann algebras, Lie groupoids, cyclic cohomology and spectral triples. The schedule can be found below.
15-1 (talk 1): Introduction & von Neumann algebras I: definitions, type classification
Speaker: Jort de Groot. Notes can be found here
22-1 (talk 2): von Neumann algebras II: modular automorphism group & crossed product
Speaker: Quirijn Boeren. Notes can be found here
29-1 (talk 3): von Neumann algebras III: modular automorphism group & crossed product (remainder last week)
Speaker: Quirijn Boeren. Notes can be found here
5-2 (talk 4): Foliations and Lie groupoids
Speaker: Mikhail Isachenkov. Notes can be found here
12-2 (talk 5): Foliations and Lie groupoids (remainder last week) -- Lie algebroids
19-2 (talk 6): Lie algebroids -- Lie groupoids
26-2 (talk 7): Cyclic cohomology I: Hochschild cohomology
Speaker: Shin Komatsu. Notes can be found here
5-3 (talk 8):Cyclic cohomology II: definitions
Speaker: Jurre Groen. Notes can be found here
12-3 (talk 9): Cyclic cohomology III: K-theory and Chern character
Speaker: Raf Bocklandt. Notes can be found here
26-3 (talk 10): Spectral triples, including example of spin manifolds
Speaker: Teun van Nuland. Notes can be found here
2-5 (talk 11): Cyclic cohomology IV: K-theory and Chern character
Speaker: Hessel Posthuma
15-4 (talk 12): Noncommutative Chern character and Fredholm modules
Speaker: Hessel Posthuma. Notes can be found here (including those of talk 11)
22-4 (talk 13): Local index theorem for spectral triples I
Speaker: Jort de Groot. Notes can be found here
7-5 (talk 14): Local index theorem for spectral triples II
Speaker: Jort de Groot. Notes can be found here
14-5 (talk 15): Index theorem for transverse foliations I
Speaker: Jurre Groen. Notes can be found here
21-5 (talk 16): Index theorem for transverse foliations II
Speaker 1: Jurre Groen. Notes can be found here
Speaker 2: Sven Holtrop.
4-6 (talk 17): Index theorem for transverse foliations III
Speaker 1: Sven Holtrop
Speaker 2: Hessel Posthuma
(For a more detailed version, see here)
1. Introduction & von Neumann algebras I: definitions, type classification [B]
2. von Neumann algebras II: modular automorphism group, crossed product [B]
3. von Neumann algebras III: factors and subfactors [B,JS]
4. Foliations and Lie groupoids [MM, K]
5. Lie algebroids and Lie’s theorems [MM, K]
6. Cyclic cohomology I: Hochschild cohomology [K, W]
7. Cyclic cohomology II: definitions [K, W]
8. Cyclic cohomology III: K-theory and Chern character [K, GVF]
9. Spectral triples: including basic example of spin manifolds [vS, GVF]
10. The local index theorem for spectral triples [vS, H, CM]
11. The index theorem for foliations [CM]
[B] Blackadar: Operator algebras, chapter III about von Neumann algebras
[JS] Jones–Sunder: Introduction to subfactors
[MM] Moerdijk–Mrcun. Introduction to foliations and Lie groups.
[K] Khalkhali: Basic Noncommutative Geometry
[H] Higson: The local index formula in Noncommutative Geometry
[vS] Van Suijlekom: Noncommutative Geometry and Particle Physics
[W] Weibel: Introduction to homological algebra
[GVF] Gracia-Bondia, Varilly and Figueroa: Elements of Noncommutative Geometry
[CM] Connes and Moscovici: The local index formula in noncommutative geometry