## The Atiyah—Singer Index Theorem, Revisited From Supersymmetry

## Supersymmetric Quantum Field Theory

## Estimations of major and minor arcs

## Hitchin representations and higher rank lattices

Let S be a closed orientable surface of genus at least 2. The Hitchin component is a special connected component of the space of representations of the fundamental group of S in SL(n,R). All the representations in it are faithful and have discrete image. Motivated by the study of higher rank lattices, the goal of this talk is to construct Hitchin representations whose image is contained in a lattice of SL(n,R). This gives new examples of Zariski-dense subgroups in arithmetic groups (these are "thin" subgroups). I will start by presenting the Hitchin component and motivating the problem. I will then explain the construction that relies on arithmetic considerations.

## Frobenius theorem in Algebraic Geometry

In the past century, foliations have been studied intensively in the context of differential geometry. However, they have recently received a surge of interest from algebraic geometers as a tool to tackle long standing conjectures such as the abundance and the Green--Griffiths conjecture. The aim of this talk is to introduce foliations using the language of stacks and to interpret Frobenius theorem as a local structure theorem for these stacks.

## Crash course in differential topology, part II

## Crash course in differential topology, part I

## Contact surgery

Contact 3-manifolds are smooth 3-manifolds which carry a special geometric structure. A way to get new contact manifold is via contact surgery. One could think of this procedure as an analogue of Dehn surgery in the smooth world.

In this talk, I'll briefly define contact surgery and its applications. I'll also mention some of the open problems which I find interesting. Next I'll talk about contact surgery numbers. Part of this will be joint work with Marc Kegel.

## A friendly introduction to Waring's problem and the circle method

## Fundamental groups of affine manifolds

The study of the fundamental group of an affine manifold has a long history that goes

back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact

Euclidian affine manifold has a subgroup of a finite index such that every element of this

subgroup is translation. The motivation was the study of the symmetry groups of crys-

talline structures which are of fundamental importance in the science of crystallography.

A natural way to generalize the classical problem is to broaden the class of allowed mo-

tions and consider groups of affine transformations. In 1964, L. Auslander in his paper

”The structure of complete locally affine manifolds” stated the following conjecture, now

known as the Auslander conjecture: The fundamental group of a compact complete locally

flat affine manifold is virtually solvable.

In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”,

J.Milnor asked if a free group can be the fundamental group of complete affine flat mani-

fold.

The purpose of the talk is to recall the old and to talk about our new results in light of

these questions. Our approach is based on the study of the dynamic of an affine action.

The talk is aimed at a wide audience and all notions will be explained.

## On the denominators of the special values of the partial zeta functions of real quadratic fields

It is classically known that the special values of the partial zeta functions of real quadratic fields, or more generally, of totally real fields at negative integers are rational numbers.

In this talk, I would like to discuss the denominators of these rational numbers in the case of real quadratic fields.

More precisely, Duke recently presented a conjecture which gives a universal upper bound for the denominators of these special values of the partial zeta functions of real quadratic fields.

I would like to explain that by using Harder's theory on the denominator of the Eisenstein class for SL(2,Z), we can prove the conjecture of Duke and moreover the sharpness of his upper

bound. This is a joint work with Ryotaro Sakamoto.

## Van Est maps and the perturbation lemma. Part II

This talk is intended to be an educational lecture to learn about the Perturbation Lemma and the Van Est maps. The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. We will revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we will obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we will construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level.

## Supersymmetric Quantum Mechanics, Part 2

## Cobordism categories of manifolds with disc-presheaf structures

The study of cobordism categories of manifolds has had important consequences on the topology of manifolds, in particular regarding the cohomology

of moduli spaces of manifolds in certain stable ranges. An interesting variant are embedded cobordism categories, where we require all closed manifolds

be embedded in a background manifold W, and where cobordisms are taken in W \times I. Using ideas from embedding calculus, we propose a notion of

a cobordism category of manifolds with certain disc-presheaf structures, which generalizes the classical cobordism category, and conjecturally recovers

the homotopy type of embedded cobordism categories.

## Schur indices and their modular properties

In this talk, we will introduce 4d superconformal index, which is a type of Witten index, when taking the Schur limit, it is called Schur index

and has nice mathematical properties. We study the case of unflavored Schur index for the BCD-type gauge groups by two methods and

discover some interesting new features.

## tba

## Violation of local-to-global principles for rationality and linearizability

In the first part, based on the preprint arXiv:2305.03481 (to appear in Comptes Rendus Math.), we show that even within a class of varieties where the Brauer–Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base change invariant form, may be insufficient for explaining counter-examples to the local-to-global principle for rationality. We exhibit examples of toric varieties and rational surfaces over an arbitrary global field $k$ each of those, in the absence of the Brauer obstruction, is rational over all completions of $k$ but is not $k$-rational.

In the second part, based on a work in progress (in collaboration with Jean-Louis Colliot-Thélène), for every global field $k$ and every $n \ge 3$ we give an example of a birational involution of $\mathbb P^n_k$ (that is an element $g$ of order $2$ in the Cremona group $\mathrm{Cr}(n, k))$ such that $\bullet$ $g$ is not linearizable; $\bullet$ $g$ is linearizable in all $\mathrm{Cr}(n, k_v)$.

## Pasting diagrams beyond acyclicity

Many of the available algebro-combinatorial frameworks for presenting pasting diagrams, or more general diagrams in n-categories, rely on a global acyclicity

condition to ensure that a "combinatorial diagram" is equivalent to the n-functor that it presents.

This is somewhat inconvenient, as global properties tend to be unstable, and many diagrams that arise in practice are not acyclic.

In my talk, I would like to give an overview of the combinatorics of higher-dimensional diagrams when (global) acyclicity is relaxed to (local) "regularity",

a condition topological in nature: what works equally well, what works better, and what still requires some milder form of acyclicity.

## Knots in Contact Structures

A contact structure on an odd-dimensional manifold is a maximally non-integrable hyperplane field that vanishes nowhere. In dimension three, this structure distinguishes two special classes of knots: Legendrian knots and transverse knots. In this talk, I will discuss problems related to Legendrian and transverse knots.

## Equivariant algebraic K-theory and Artin L-functions

The Quillen-Lichtenbaum Conjecture, proved by Voevodsky-Rost, originally states that special values of the Dedekind zeta function of a number field are computed by sizes of its algebraic K-groups. In this talk, I will sketch how to generalize this conjecture to Artin L-functions of Galois representations by considering equivariant algebraic K-groups with coefficients in those representations. This is joint work in progress with Elden Elmanto.

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