19223811 Forschungsmodul: Topologie "Cyclic Homology"
- FU-Students should register via Campus Management.
- Non-FU-students should register via MyCampus/Whiteboard.
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.
|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||Paul Brommer-Wierig|
|16.11.||Talk 5: Connes cyclic category||Ibrahim El Agami|
|23.11.||Talk 6: Tor and Ext interpretation of HC||Vittorio di Fraia|
|30.11.||Talk 7: Crossed simplicial groups||Aldo Kiem|
|07.12.||Talk 8: Cyclic spaces||John Maar|
|14.12.||Talk 9: HC and S^1-equivariant homology||Lucas Piessevaux|
|04.01.||Talk 10: Examples of cyclic sets||Evgeniya Lagoda|
|11.01.||Talk 11: Free loop spaces||Kyle Huang|
|18.01.||Talk 12: HC of group algebras||Manuel Staiger|
|25.01.||Talk 13: Classical chern character||João Tavares|
|01.02.||Talk 14: Generalized chern character||Prof. Dr. Elmar Vogt|
|08.02.||Talk 15: The Dennis trace||Prof. Dr. Elmar Vogt|
- 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.