# 2024 Graduate Summer School Course Descriptions

__Week 1 (July 7 - 13)__

__Week 2 (July 14 - 20)__

__Week 3 (July 21 - 27)__

__Joseph Ayoub, Universität Zürich__

Title: A^1-algebraic topology (following F. Morel)

Abstract: The goal of the lectures is to give an introduction to the subject of unstable motivic homotopy theory. In particular, we will introduce the Morel-Voevodsky category of motivic spaces, sketch a proof of Morel's fundamental theorems on the homotopy sheaves of motivic spaces, and present Morel's computation of the first unstable homotopy sheaves of motivic spheres. The main reference is Morel's book ``A^1-algebraic topology over a field''. No prior knowledge of the subject will be assumed, however familiarity with basic notions from algebraic geometry and algebraic topology will be very helpful.

__Hélène Esnault, Freie Universität Berlin, Harvard University, and the University of Copenhagen__

Title: Arithmetic properties of local systems.

Abstract: A building block of homotopy theory is the fundamental group of varieties, in topology and in arithmetic geometry. We know very little on it. One way to approach it is via local systems, that is linear representations modulo conjugation. Among those there are the ones coming from geometry, that is the motivic ones. Deep conjectures predict over various fields when local systems should but motivic.

I’ll go through a choice of classical and recent conjectures, problems and results in arithmetic geometry.

The general reference for the state of the art in the spring 2023 is the Springer Lecture Notes 2337

https://link.springer.com/book/10.1007/978-3-031-40840-3

I’ll build up on the text. Depending on the latest developments in this direction, I might add new material.

__Frederic Deglise, CNRS, ENS Lyon__

Title : Characteristic classes in stable motivic homotopy theory

Abstract : Characteristic classses is a pillar of differential geometry. The book of Milnor and Stasheff was a corner stone and the work of Quillen on formal group laws and cobordism showed their fundamental importance in stable homotopy, giving rise to entirely new subject such as chromatic homotopy theory.

It is natural to look for an analogous picture in stable motivic homotopy theory where varieties and CW-complexes are replaced by algebraic varieties and schemes. Such an extension has been actively developed, first as an essential piece of Voevodsky’s proof of the Milnor conjecture.

The talk will present these developments in stable motivic homotopy theory, from the successful extension of Quillen’s work by Levine and Morel, to emerging topics such as Panin and Walter’s orientation theories leading to quadratic enumerative geometry.

__Philippe Gille, Université Claude Bernard, Lyon 1__

Abstract : We shall present the theory of G-torsors (or G-bundles) in algebraic geometry which includes for example vector bundles and quadratic bundles (Grothendieck-Serre, 1958). In a first part, we focus on the case of an affine smooth connected curve over an algebraically closed field k; we shall show that G-torsors are trivial for a semisimple k-group G.

In a second part we will consider the case of a Dedekind ring. This includes the ring of integers Z and affine curves over an arbitrary field. We shall discuss the important cases of the affine line (Raghunathan-Ramanathan, 1984) and of the punctured affine line. This will be an opportunity to deal with étale cohomology and patching techniques.

In a third part, we will deal with the case of affine smooth surfaces and will discuss the case of the affine plane.

References:

V. Chernousov, P. Gille, A. Pianzola, Three-point Lie algebras and Grothendieck's dessins d'enfants. Math. Res. Lett. 23 (2016), 81-104.

J.-L. Colliot-Thélène, J.-J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Mathematische Annalen 244 (1979), 105-134.

N. Guo, The Grothendieck–Serre conjecture over semilocal Dedekind rings, Transform. Groups, 2022.

J.S. Milne, Lectures on etale cohomology, https://www.jmilne.org/math/CourseNotes/LEC.pdf

M. Ojanguren, R. Sridharan, Cancellation of Azumaya algebras.

J. Algebra 18(1971), 501-505.

M. S. Raghunathan, A. Ramanathan, Principal bundles on the affine line. Proc. Indian Acad. Sci. Math. Sci. 93 (1984), 137–145.

__Daniel Krashen, University of Pennsylvania__

Title: Field arithmetic and the complexity of Galois cohomology

Abstract: Galois cohomology classes arise naturally in a wide range of contexts, as invariants of algebraic objects, as obstructions in algebraic and arithmetic geometry, and more generally as tools for understanding field arithmetic. While the resolution of the Bloch-Kato conjectures and the new techniques they have provided has given us important insights into Galois cohomology, fundamental questions remain.

In these talks we will try to discuss some of the following topics:

- A brief summary of definitions and standard interpretations of Galois cohomology classes

- A survey of how various classes of algebraic structures give rise to Galois cohomology classes via cohomological invariants and in particular via connections to motives of projective homogeneous varieties

- What we know about the structure of the Galois cohomology ring and representing its elements, and a summary of the main open questions, including: period-index, symbol length, vanishing of Massey products and the relationship between Diophantine and cohomological dimensions

- A summary of what we know about some of these problems, and various techniques and perspectives, including the use of Milnor conjectures and quadratic forms, and the development of local-global principles in various contexts

__Alexander Merkurjev, University of California, Los Angeles and Federico Scavia, University of California, Los Angeles__

Title: Massey products in Galois cohomology

Abstract: A fundamental question in Galois theory is the profinite inverse Galois problem: Which profinite groups are realizable as absolute Galois groups of fields? A historically fruitful approach to the profinite inverse Galois problem has been to give constraints on the cohomology of absolute Galois groups. The most spectacular example of this is the Norm-Residue Theorem (the Bloch–Kato Conjecture), proved by Rost and Voevodsky. This theorem implies that the mod p cohomology ring of the absolute Galois groups of a field containing a primitive p-th root of unity admits a presentation with generators in degree 1 and relations in degree 2. This property is false in general for arbitrary profinite groups, and so gives a way to prove that a profinite group does not arise as the absolute Galois group of a field.

Another tool to establish constraints on the Galois cohomology of absolute Galois groups is the theory of Massey products. The Massey Vanishing Conjecture of Minac and Tan predicts that all Massey products in the Galois cohomology of a field vanish as soon as they are defined. We will give an introduction to Massey products and describe recent progress on the Massey Vanishing Conjecture.

__Sabrina Pauli, TU Darmstadt__

Title: Motivic explorations in enumerative geometry

Abstract: Motivic homotopy theory allows to do enumerative geometry over an arbitrary base which leads to additional arithmetic and geometric information. The goal of the mini-workshop is to explain why and how this works. Furthermore, I will provide a toolbox to solve questions in enumerative geometry in this set up including the use of tropical geometry.

__Burt Totaro, University of California, Los Angeles__

Title: Classifying spaces in motivic homotopy theory

Abstract: The classifying space BG of a reductive group G is a central object in motivic homotopy theory, similar to the classifying space of a compact Lie group in classical homotopy theory. The course will show how to compute with equivariant Chow groups. We will present connections with the integral Hodge conjecture, finiteness problems for Chow groups, and Hodge theory in characteristic p.

__Kirsten Wickelgren, Duke University__

Title: A1-homotopy theory and the Weil conjectures

Abstract: These lectures will introduce the A1-derived category and related notions. We will construct the cellular homology of Morel and Sawant, and analogues of part of the Weil conjectures in A1-homotopy theory. The new material is joint with Tom Bachmann, Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.

References:

M. Bilu, W. Ho, P. Srinivasan, I. Vogt, K.W. "Quadratic enrichment of the logarithmic derivative of the zeta function"

M. Hoyois "A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula"

J.S. Milne "Étale cohomology"

F. Morel and A. Sawant "Cellular A1-homoloty and the motivic version of Matsumoto's theorem"

F. Morel "A1-algebraic topology over a field"