Motives Seminar-WS-2017/18

Topological cyclic homology following Nikolaus-Scholze

Tues., 14-16, WSC-S-U-3.02

This semester will be going through the recent paper of Nikolaus and Scholze on a new construction of topological cyclic homology. As a preparation, we will first give a quick overview of the theory of infinity categories.

Program

Part I: Introduction to infinity categories

Lecture 1. (17.10) Lorenzo Mantovani. The language of $\infty$-categories
Lecture 2. (24.10) Alejo Avila-Lopez. Presentable $\infty$-categories
Lecture 3. (7.11) Enzo Serandon. Symmetric monoidal $\infty$-categories
Lecture 4. (14.11) Maria Yakeson. Stable $\infty$-categories

Part II: Topological cyclic homology

Lecture 1. (Nov. 21). Marc Levine. Overview of the Nikolaus-Scholze paper [4].
Lecture 2.(Nov. 28). Gabriela Guzman. The Tate construction.
Lecture 3. (Dec. 5). Ran Azouri. Multiplicativity of the Tate construction I.
Lecture 4. (Dec. 12). Rakesh Pawar. Multiplicativity of the Tate construction II.
Lecture 5. (Dec. 19). Tom Bachmann. Genuine equivariant homotopy theory.
Lectures 6-7. (Jan. 9, Jan. 16). Alejo Lopez-Avila. $(p)$-cyclotomic spectra and comparison results.
Lecture 8. (Jan. 23). Daniel Harrer. The Tate diagonal.
Lectures 9-10. (Jan. 30, Feb. 6). Sasha Samokhin. The cyclotomic structure on THH.

Bibliography

[1] Marc Hoyois. Equivariant classifying spaces and cdh descent for the homotopy K-theory of tame stacks. arXiv:1604.06410.

[2] Jacob Lurie. Higher topos theory. Number 170. Princeton University Press, 2009.

[3] Jacob Lurie. Higher algebra, May 2016.

[4] Thomas Nikolaus, Peter Scholze. On topological cyclic homology. arXiv:1707.01799.

[5] Thomas Nikolaus. Stable $\infty$-operads and the multiplicative Yoneda lemma. arXiv:1608.02901.

Resources

Program for Part I: Infinity categories

Program for Part II: Topological cyclic homology

Federico Binda's notes from the Haramura workshop on topological cyclic homology (29 MB).