The lectures on elliptic genera are Wednesdays 14:14-15:30 in room CM 1 106 (EPF Lausanne) starting Sept. 17. More informations
Slides on elliptic genera
Material covered in the lectures
- September 17: survey on elliptic genera
- September 24: vector bundles, Grassmann manifolds, universal bundles (see [MiSt], § 2, 3, 5)
- October 1: NO LECTURE !!!
- October 8: Chern classes and Pontrjagin classes (see [MiSt], § 9, 14)
- October 15: Pontrjagin numbers, Pontrjagin numbers of complex projective space, oriented bordism ring, signature (see [MiSt], § 14, 16, 17 and [Hi], § 8)
- October 22: Pontrjagin-Thom construction, multiplicative sequences, genera, Hirzebruch's signature theorem
- October 29: Milnor's exotic spheres, signature operator, elliptic operators ... (see [AtSi], [MiSt] § 19, 20 and [HiBeJu], Chapter 5)
- November 5: Topological K-theory and the Atiyah-Singer index theorem (see [AtSi] and [HiBeJu], Chapter 5)
- November 12: Discussion of the index theorem, examples (de Rham complex, signature complex), localization in K-theory (see [AtSi], [AtSe] and [HiBeJu], Chapter 5)
- November 19: Lefschetz fixed point formula (LFF) (see [AtSi], [AtSe] and [HiBeJu], Chapter 5)
- November 26: Applications of the LFF to the de Rham complex, signature operator, Dirac operator, A-hat vanishing theorem (see [AtSi], [AtSe], [HiBeJu], Chapter 5, [AtHi], [LaMi])
- December 3: Clifford algebras, Dirac operator, relation to positive scalar curvature, Elliptic genera (see [LaMi], [HiBeJu])
- December 10: Introductio to Signature of the free loop space (elliptic genus) and the Witten genus ([HiBeJu])
- December 17: Proof of the vanishing theorem for the Witten genus for String manifolds with symmetry (see [De])
[AtHi]: Atiyah, Hirzebruch : Spin manifolds and group actions, in: Essays in Topology and Related Subjects, Springer (1970), pp. 18-28
[AtSe]: Atiyah, Segal: The Index of Elliptic Operators II, Ann. Math. 87, 531-545 (1968)
[AtSi]: Atiyah, Singer: The Index of Elliptic Operators I and III, Ann. Math. 87, 484-530 and 546-604 (1968)
[De]: Dessai: Spin^c-Manifolds with Pin(2)-Action,
Math. Ann. 315, 511-528 (1999)
[Hi]: Hirzebruch: Topological methods in algebraic topology, Springer
[HiBeJu]: Hirzebruch, Berger, Jung: Manifolds and modular forms, Vieweg
[LaMi]: Lawson, Michelson: Spin Geometry, Princeton University Press
[MiSt]: Milnor, Stasheff: Characteristic classes, Princeton
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