# Abstracts and video

## Summer School (Downloadable PDF):

### Lecture Series:

**Frédéric Déglise:*** Six functors formalism in motivic homotopy theory*

Video: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5

One of the main legacy of Grothendieck's mathematical work is the six functors formalism, whose initial goal was to provide a framework for duality results in sheaf theory but that has largely superseeded this goal.

After the work of Voevodsky and Ayoub, we now have at our disposal a general machinery to produce six functors formalisms, especially adapted to Voevodsky's motivic categories. In the first part of this lecture series I will recall Voevodsky's axiomatic of cross functors, the results of Ayoub and complements that Cisinski and I have brought to the theory. We will see some of the main examples of categories that fulfill this axiomatic, based on various tools: simplicial sheaves, derived categories, relative cycles and modules over ring spectra. In the second part, I will go into more refined properties of these categories which ultimately lead to duality theorems. Main themes here are cohomological descent, absolute purity and constructiblity. The last part will be concerned with functoriality of motivic categories, which encompass comparison theorems and realization functors.

**Oliver Röndigs:** *An introduction to motivic homotopy theory*

Video: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5

This lecture series will constitute an introduction to motivic homotopy theory, with a view towards some computations. After providing the basic framework, prominent examples of generalized motivic cohomology theories will be discussed via their representing spectra. Some methods of computation will be illustrated with these examples.

**Nikita Semenov:*** Applications of Chow motives to algebraic groups*

Video: Lecture 1, Lecture 2, Lecture 3, Lecture 4

The concept of the motive is fundamental in the modern algebraic geometry. Originally it was introduced by Alexander Grothendieck as an attempt to unify cohomology theories. Meanwhile the motives became one of the main languages in the algebraic geometry to formulate and to solve its problems. Moreover, the theory of motives has a much broader impact reaching into algebraic number theory and representation theory.

In the lectures I will introduce the category of Grothendieck's Chow motives and will focus on applications of motives to the theory of algebraic groups. In particular, I plan to discuss Hoffmann's conjecture on the Witt indices of quadratic forms (proven by Karpenko), relations between motives and cohomological invariants of algebraic groups, and, if time permits, recent developments, related to computations of the Chow groups of quadrics, where the Morava K-theory plays a crucial role.

**Matthias Wendt: ***Motivic cohomology, group homology and scissors congruences*

Video: Lecture 1, Lecture 2, Lecture 3, Lecture 4

There are different ways to view rational motivic cohomology of fields: originally conceived as weight spaces in rational K-theory, they can be viewed as arising from the group homology of the infinite general linear group, or from more modern viewpoints as arising from algebraic cycles in affine spaces via Bloch's cycle complexes. Further (yet conjectural) views on rational motivic cohomology have been developed, in particular by Bloch and Goncharov: Dehn complexes arising from scissors congruence questions related to Hilbert's third problem and polylogarithm complexes related to explicit formulas for regulators on K-theory.

The goal of the lectures is to give an introduction to these various viewpoints on rational motivic cohomology and their relations. The most complete picture was obtained in the case of indecomposable $K_3$ by work of Bloch, Suslin, Dupont and Sah (and I'll try to explain this case in detail). There are various strong conjectures relating these approaches, e.g. Suslin's rank conjecture relating the rank filtration from group homology to the weight filtration from the cycle complexes, or Goncharov's conjectures on the relation between motivic cohomology of the complex numbers and Hilbert's third problem. I'll discuss some of these conjectures, what is currently known about them, and how motivic conjectures relate to open questions around Hilbert's third problem.

### Short talks:

**Håkon Kolderup:** *Cohomological correspondence categories*

Since Suslin and Voevodsky’s introduction of finite correspondences, several alternate correspondence categories have been constructed in order to provide different linear approximations to the motivic stable homotopy category. In joint work with Andrei Druzhinin, we provide an axiomatic approach to a class of correspondence categories that are defined by an underlying cohomology theory. For such cohomological correspondence categories, one can prove strict homotopy invariance and cancellation properties, resulting in a well behaved associated derived category of motives.

**Jonas Irgens Kylling:** *Slice spectral sequence calculations of hermitian K-theory and Milnor’s conjecture on quadratic forms for rings of integers*

The slice filtration was introduced by Voevodsky as a motivic analogue to the Postnikov tower. We will outline how to use the slice spectral sequence to calculate hermitian K-theory of rings of integers. We prove a variant of Milnor’s conjecture on quadratic forms for rings of integers and relate hermitian K-theory to special values of zeta-functions. This is joint work with Oliver Röndigs and Paul Arne Østvær.

**Pavel Sechin:** *Morava K-theories: Chern classes, the gamma filtration and applications to projective homogeneous varieties*

Morava K-theories are oriented cohomology theories obtained from algebraic cobordism of Levine-Morel by change of coefficients. Motives corresponding to them provide a linearly ordered series of obstructions for the splitting of the Chow motive. I will explain the tools which help to gain information on Chow groups using splitting of Morava motives, and will briefly discuss known such cases among homogeneous varieties. The latter part is based on the joint work with Nikita Semenov.

**Maria Yakerson:** *Framed transfers in motivic homotopy theory*

In motivic settings, generalized cohomology theories acquire additional structure called framed transfers. We will discuss this structure from different perspectives. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.