Workshop: Arithmetic Geometry and Topological Cyclic Homology
University of Copenhagen,
18-20 April 2016

# Schedule (AUD 4)

## Monday 18 April

10:20 Alain Connes
The scaling site and Riemann-Roch theorem of type II - part 1
12:30-2:30 Lunch break
2:30 Lars Hesselholt
Periodic topological cyclic homology and the Hasse-Witt zeta function
4:00-5:30

Alain Connes
The scaling site and Riemann-Roch theorem of type II - part 2

## Tuesday 19 April

10:20 Bjørn Dundas
Some calculations of iterated Hochschild homology
12:30 Lunch break
1:15-2:15 André Joyal
A classifying topos for Penrose tilings
2:45-3:15 Tea
3:15-4:15 Alain Connes - Harald Bohr Lecture
Geometry and the Quantum

## Wednesday 20 April

9:30 James Borger
Witt vectors of semi-rings
12:30-2:30 Lunch break
2:30-4:00 Discussions

# Abstracts

James Borger: Witt vectors of semi-rings
I'll explain a definition of the big Witt vectors that does not use subtraction. This allows us to extend the functor to the category of semirings, in fact not just as a functor but as a representable comonad. I'll then discuss in some detail the Witt vectors of the semirings of natural numbers and positive reals. I'll then discuss some ways these constructions relate to some objects, possibly hypothetical, of arithmetic interest such as zeta functions, Deninger's spaces, and the arithmetic square.

Alain Connes: Geometry and the Quantum
I will start from Riemann’s question, in his inaugural lecture on the foundation of geometry, on the validity of his formalism in the infinitely small. I will explain how the new paradigm of noncommutative geometry, based on the quantum formalism, gives a very precise answer to Riemann's query, and will show how the new spectral paradigm leads to a complete geometric understanding of the Standard model of particle physics coupled with gravity.

Bjørn Dundas: Some calculations of iterated Hochschild homology
I will present some concrete calculations. Many of these calculations yield answers we’ve seen before in other contexts. One of the more provocative is the twice iterated Hochschild homology of the integers which turns out to be isomorphic to topological Hochschild COhomology. I don’t know why; perhaps you can tell me.
Some calculations are now quite elementary. For instance, the iterated topological Hochschild homology of F_p, is directly accessible through the fact that the suspension of the torus splits, a spectral sequence of Greenlees, and manipulations very similar to Cartan and Serre’s calculations of the cohomology of Eilenberg MacLane spaces. This implies that the fixed points can detect periodic classes on any desired chromatic level. This calculational scheme of “desuspending the suspended torus” does not work for all ring spectra, and in some way depends on smash powers of the Eilenberg MacLane spectrum of F_p beeing “smooth enough” to unstably trivialize the attaching maps in the cell decomposition of the torus.
The talk contains joint work with Ausoni, Lindenstrauss and Richter.

Lars Hesselholt: Periodic topological cyclic homology and the Hasse-Weil zeta function
We propose a definition of periodic topological cyclic homology and show that, for schemes smooth and proper over a finite field, the infinite dimensional cohomology theory that results provides a natural vessel for Deninger’s cohomological interpretation of the Hasse-Weil zeta function by regularized determinants. In this way, the theory may be seen as a non-archimedean analogue of the cohomological interpretation of Serre's archimedean local L-factors in terms of cyclic homology given by Connes and Consani.

André Joyal: A classifying topos for Penrose tilings
In his book on non-commutative geometry, Alain Connes observes that the "space" of Penrose tilings is "non-commutative" with an associated C-star algebra. We will show that the notion of (Penrose) tiling is geometric (in the topos-theoretic sense). It follows that Penrose tilings have a classifying topos. This is consistent with Connes's philosophy that toposes are non-commutative spaces.

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