This small workshop is organized around the visit of Lars Hesselholt and Thomas
Nikolaus. It is financed by the ANR-project Chromatic
homotopy and K-Theory and the LAGA.

Introductory talks on THH are scheduled on Wednesday afternoon, before more specialized lectures by Thomas Nikolaus on Thursday morning and Lars Hesselholt on Thursday afternoon.## Practical Information

**Place:**

LAGA - Institut Galilée

Université Paris 13

99 avenue Jean Baptiste Clément

93430 Villetaneuse

The talks will be in the Institut Galilée. Here is a description on how to come to the LAGA.

**Registration:**

No registration necessary. Please contact Christian Ausoni if you have any questions or want to join in for lunch on Thursday.
## Schedule

### Wednesday 18th of January 2017 (Amphi Copernic, Institut Galilée)

### Thursday 19 January 2017 (Amphi Euler, Institut Galilée)

*
*

Introductory talks on THH are scheduled on Wednesday afternoon, before more specialized lectures by Thomas Nikolaus on Thursday morning and Lars Hesselholt on Thursday afternoon.

LAGA - Institut Galilée

Université Paris 13

99 avenue Jean Baptiste Clément

93430 Villetaneuse

The talks will be in the Institut Galilée. Here is a description on how to come to the LAGA.

No registration necessary. Please contact Christian Ausoni if you have any questions or want to join in for lunch on Thursday.

**14:00-14:45**

Bruno Stonek (Paris 13),*Introduction to (topological) Hochschild Homology*.

We define Hochschild homology and topological Hochschild homology as a certain derived tensor product, highlighting how it is the same construction but carried either in a category of chain complexes or in a category of spectra. We give a relationship between them. Then, we give an alternative definition via the cyclic bar construction. We sketch a computation of THH(KU), where KU is the periodic complex topological K-theory spectrum.**14:45-15:30**

Christian Ausoni (Paris 13),*A sample of older and newer computations of THH*.

Topological Hochschild has been computed for many spectra using a large range of techniques. In this lecture, I will sketch several examples of computations, including the Eilenberg-Mac Lane specrum of a finite field, and the second Johnson-Wilson spectrum (joint work with Birgit Richter).**16:00-16:45**

Eva Höning (Paris 13),*The topological Hochschild homology of algebraic K-theory of finite fields*.

Angeltveit and Rognes calculated the mod p homology of THH(K(F_q)) via the Bökstedt spectral sequence, when q is a topological generator of the p-adic units. We mention when their methods apply to further cases and when other methods are necessary. I explain how a spectral sequence of M. Brun for the calculation of THH of rings generalizes to S-algebras. We show how this spectral sequence can be used for an alternative computation of V(1)_* THH(ku_p), where ku_p is p-completed connected complex K-theory. Finally, we apply the spectral sequence for K(F_q) and explain first results.**16:45-17:30**

Geoffroy Horel (Paris 13),*Introduction to factorization homology*.

I will explain the construction of factorization homology. It is a pairing between d-dimensional manifolds and algebras over the little d-disks operad. When d=1, we obtain a model for THH that makes the circle action more transparent than the standard model. I will try to explain the proof of this fact (with more or less details depending on the time I have).

**9h15-10h15**and**10:30-11:45**

Thomas Nikolaus (MPIM Bonn),*On topological cyclic homology*.

In the first part of the talk I will give an overview of the result of joint work with P. Scholze. We establish a new model for the cyclotomic structure on topological Hochschild homology and in particular a resulting simplification of the formula for topological cyclic homology in relatively basic terms (the Tate spectrum will come up, so I will also explain that). Finally we indicate basic applications and results.

In the second half we explain how to prove these results, in particular the computational key lemma (which we call the Tate orbit lemma). Another key to understand the cyclotomic structure will be a generalization of the Segal conjecture which is in the finite type case due to Rognes-Nielsen and whose generalization to the bounded below case we will explain. If time permits we will also indicate how to construct the cyclotomic trace and a refinement through "K-theory of endomorphisms".**14h00-15h00**and**15h30-16h30 (Colloquium du LAGA)**

Lars Hesselholt (Nagoya/Copenhagen),*Topological Hochschild homology and the Hasse-Weil zeta function*.

In the nineties, Deninger gave a detailed description of a conjectural cohomological interpretation of the (completed) Hasse-Weil zeta function of a regular scheme proper over the ring of rational integers. He envisioned the cohomology theory to take values in countably infinite dimensional complex vector spaces and the zeta function to emerge as the regularized determinant of the infinitesimal generator of a Frobenius flow. In this talk, I will explain that for a scheme smooth and proper over a finite field, the desired cohomology theory naturally appears from the Tate cohomology of the action by the circle group on the topological Hochschild homology of the scheme in question.**16:30**Thé du LAGA.