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19223811 Forschungsmodul: Topologie "Cyclic Homology"

Winter Term 2021/2022

Dozenten: Dr. Gabriel Angelini-Knoll, Prof. Dr. Elmar Vogt


  • Time and place:  Tuesday,  4pm -- 6pm, SR 009, Arnimallee 6.

  • Leistungsnachweis/criteria for proof of performance:
    Grade and credit points will be awarded based on a presentation and written summary.


Prerequisites: We assume basic knowledge of topology as taught in Topology I and II. We also assume basic knowledge of homological algebra as taught in Topology II or an algebra course.
This seminar will likely also be of interest to students studying algebra in addition to topology students.

Content: The seminar will cover advanced topics from algebra and topology.

Cyclic homology has its roots in the work of Alain Connes on non-commutative de Rham cohomology and work of Loday on algebraic K-theory. Cyclic homology also recovers the homology of a free loop space and homology of spaces with an action of the circle group. The field of cyclic homology and applications to algebraic K-theory remains an active area of research including the important recent groundbreaking work of Thomas Nikolaus and Peter Scholze, which has led to several new computational advances in the field.
In this seminar, we will discuss the algebraic constructions of Hochschild homology and cyclic homology and the Hochschild--Konstant--Rosenberg theorem. In the remaining time, we plan to discuss topological applications culminating in a discussion of generalized Chern characters. The material will likely be of interest to algebraists as well as topologists.
The primary reference is Loday's book "Cyclic homology." We will supplement this material with Weibel's account in "An Introduction to Homological Algebra" and other references as needed.

Talks

DateTitleSpeaker
19.10. Talk 1: Organization and overview Dr. Gabriel Angelini-Knoll
26.10. Talk 2: Hochschild homology Dr. Gabriel Angelini-Knoll
02.11. Talk 3: Cyclic homology Dr. Gabriel Angelini-Knoll
09.11. Talk 4: The HKR Theorem N.N.
16.11. Talk 5: Connes cyclic category N.N.
23.11. Talk 6: Tor and Ext interpretation of HC N.N.
30.11. Talk 7: Crossed simplicial groups N.N.
07.12. Talk 8: Cyclic spaces N.N.
14.12. Talk 9: HC and S^1-equivariant homology N.N.
04.01. Talk 10: Examples of cyclic sets N.N.
11.01. Talk 11: Free loop spaces N.N.
18.01. Talk 12: HC of group algebras N.N.
25.01. Talk 13: Classical chern character N.N.
01.02. Talk 14: Generalized chern character N.N.
08.02. Talk 15: The Dennis trace N.N.

Literature:

  • Jean-Louis Loday: Cyclic homology. Second edition. Grundlehren der Mathematischen Wissenschaften, 301. Springer-Verlag, Berlin, 1988.
  • Charles A. Weibel: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge 1994.