Seminar on Spin Geometry and Applications

Chair: Luca Asselle

12.10.2023. Jan Eyll  "Clifford algebras."
Abstract: Review of quadratic spaces (characteristic different from two), Clifford algebras, Z2-grading on Clifford algebras, description of linear structure of Clifford algebras, center of Clifford algebras, Clifford algebras of direct sumsof quadratic spaces, Clifford algebras and complexifications. References: Sections 1.1, 1.2, and 1.3 (up to the first proposition) of Friedrich’s book.


                             

19.10.2023. Michael Vogel "Classification of Clifford algebras and their re- presentations."
Abstract: Clifford algebras of standard non-degenerate (real or complex) quadratic spaces, explicit identifications in low dimensions, periodicity results, description of these Clifford algebras via matrix algebras, irreducible representations of these Clifford algebras, spinor representations. References: Section 1.3 of Friedrich’s book.


                             

26.10.2023. Mehmet Furkan Cosgun "The groups Spin(n) and the spin representations."
Abstract: The groups Pin(n) and Spin(n), Spin(n) as universal covering space of SO(n)for n ≥ 3, description of Lie algebra of Spin(n), spin representations of Spin(n), outlook on Spin(r, s) and the more general Spin(V, q) associated to a quadratic space (V, q), the C groups Spin (n). References: Section 1.4 to 1.6 of Friedrich’s book.


                             

2.11.2023. Jacobus Sander de Pooter "Topological K-theory."
Abstract: Classification of vector bundles, stable isomorphism classes of vector bundles, real and complex topolo- gical K-theory, Bott periodicity theorem, K-theory as a generalized cohomology theory.   References: Hatcher’s book.


                             

9.11.2023. Lukas Dahinden "The Atiyah-Bott-Shapiro construction."
Absteact: Alternative approach to K-theory via finite long exact sequences of vector bundles, Euler characte- ristics, the Atiyah-Bott-Shapiro isomorphisms relating Clifford modules to KO∗ and K∗, some explicit descriptions of generators in KO∗ in terms of Clifford modules. References: Section 1.9 of Lawson and Mitchelshon’s book.


                             

16.11.2023. Paul Anton Wilke "Principal fibre bundles and connections."
Abstract: Definition of principal fibre bundles, description via non-abelian Cech cocycles, examples, construction of associated fibre bundles, illustration via frame bundles and associated vector bundles, connections, reduction of structure groups, orientability and reduction to SO(n), Whitney sums and reduction of structure group, reduction to trivial group. References: Appendices B.1 and B.3 of Friedrich’s book.


                             

23.11.2023. Giorgia Testolina "Spin structures."
Abstract: Spin structures on vector bundles, Ccriteria for existence in terms of Stiefel-Whitney classes, relation to orientability, Spin (n) structures, examples of spin manifolds, example of (orientable!) manifolds which do no admit a spin structure, spinor bundles, associated bundles. References: Chapter 2 of Fried- rich’s book.



                             

30.11.2023. Luca Asselle  "Application I: Floer homology of cotangent bundles."
Abstract: The Abbondandolo-Schwarz isomorphism between the Floer homology of cotangent bundles and the singular homology of the free loop space of loops in the base manifold.


                             

7.12.2023. Pierre Alexandre Arlove "Application I: Floer homology of cotangent bundles, Part II."
Abstract: Second proof (a ́ la Weber-Salamon) of the isomorphism between the Floer homology of cotangent bundles and the singular homology of the free loop space via the heat flow.


                             

18.01.2024. Souheib Allout "Dirac operators."
Abstract: Connections on spinor bundles, Dirac and Dirac-Laplace operators, Lichnerowicz formula. References: Chapter 3 of Friedrich’s book.


                             

25.01.2024. Luca Asselle "Analytical properties of Dirac operators."
Abstract: Essential selfadjointness on complete manifolds, compact resolvent on compact manifolds, Fredholm properties on compact manifolds. References: Chapter 4 of Friedrich’s book.


                             

1.02.2024. Giulio Sanzeni "Application II: the Dirac equation."
Abstract: Elements of Spe- cial Relativity, spin notion in Quantum Mechanics. Derivation of the Dirac equation and its physical implications.



                             



 

The basic references for the seminar are:

  • T. Friedrich, Dirac operators in Riemannian Geometry, Graduate Studies in Mathe- matics, AMS, 2000.

  • A. Hatcher, Vector bundles and K-theory, available online at https://pi.math.cornell/ hatcher/VBKT/VBpage.html

  • H. Lawson and M. Michelshon, Spin Geometry, Princeton mathematical series, Prin- ceton University Press, 1989.

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