### Height pairings, torsion points, and dynamics

Holly Krieger
We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of elliptic curves.

### Stably irrational hypersurfaces of small slopes

Stefan Schreieder
We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension $n > 2$ and degree at least $log_2 (n) + 2$ is not stably rational. This significantly improves earlier results of Kollár and Totaro. As a byproduct of our proof, we obtain new counterexamples to the integral Hodge conjecture, answering a question of Voisin and Colliot-Thélène – Voisin.

### The compressed annotation matrix: an efficient data structure for persistent cohomology

Clément Maria
The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substancially the amount of matrix operations....

### Around Jouanolou-type theorems

Rahim Moosa
In the mid-90’s, generalising a theorem of Jouanolou, Hrushovski proved that if a D-variety over the constant field C has no non-constant D-rational functions to C, then it has only finitely many D-subvarieties of codimension one. This theorem has analogues in other geometric contexts where model theory plays a role: in complex analytic geometry where it is an old theorem of Krasnov, in algebraic dynamics where it is a theorem of Bell-Rogalski-Sierra, and in meromorphic...

### Forcing theory for transverse trajectories of surface homeomorphisms - Part 1

Patrice Le Calvez
Several recent papers on surface dynamics have used transverse foliations and maximal isotopies for homeomorphisms isotopic to the identity as a main tool in their work. In this mini-course we will introduce the basic concepts behind this tool and show a new way o deriving useful dynamical information by means of a forcing procedure. The applications involve ways of obtaining existence of non-contractible periodic points with consequences for rotation sets of toral homeomorphisms, exponential growth...

### Counting curves of given type, revisited

Juan Souto
Mirzakhani wrote two papers studying the asymptotic behaviour of the number of curves of a given type (simple or not) and with length at most $L$. In this talk I will explain a new independent proof of Mirzakhani’s results. This is joint work with Viveka Erlandsson.

### Valorisation des données de la recherche : l’offre de service de l’Inist-CNRS

Bernard Sampité
Plans de gestion des données (DMP) OPIDoR (DMP OPIdoR, Cat opidOr, Datacite) et Doranum

### Torsion groups do not act on 2-dimensional CAT(0) complexes

Piotr Przytycki
We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then the action is trivial. In particular, all actions of finitely generated torsion groups on such complexes are trivial. As an ingredient, we prove that the image of an immersed loop in a graph of girth 2π with length not commensurable to π has diameter > π. This is related to...

### Calogero-Moser cellular characters: the smooth case

Cédric Bonnafé
Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group $W$. Whenever $W$ is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is smooth, then all the Calogero-Moser cellular characters are irreducible. This implies in particular that our conjecture...

### Le problème Graph Motif - Partie 1

Guillaume Fertin
Le problème Graph Motif est défini comme suit : étant donné un graphe sommet colorié G=(V,E) et un multi-ensemble M de couleurs, déterminer s'il existe une occurrence de M dans G, c'est-à-dire un sous ensemble V' de V tel que (1) le multi-ensemble des couleurs de V' correspond à M, (2) le sous-graphe G' induit par V' est connexe. Ce problème a été introduit, il y a un peu plus de 10 ans, dans le...

### Dynamics on quotients of SL(2,C) by discrete subgroups - Lecture 2

Barbara Schapira
We will discuss old and recent results on topological and measurable dynamics of diagonal and unipotent flows on frame bundles and unit tangent bundles over hyperbolic manifolds. The first lectures will be a good introduction to the subject for young researchers.

### Modeling the chemotherapy-induced selection of drug-resistant traits during tumor growth

Doron Levy
The emergence of drug-resistance is a major challenge in chemotherapy. In this talk we will present our recent mathematical models for describing the dynamics of drug-resistance in solid tumors. Our models follow the dynamics of the tumor, assuming that the cancer cell population depends on a phenotype variable that corresponds to the resistance level to a cytotoxic drug. We incorporate the dynamics of nutrients and two different types of drugs: a cytotoxic drug, which directly...

### Isoperimetry with density

Frank Morgan
In 2015 Chambers proved the Log-convex Density Conjecture, which says that for a radial density f on $R^n$, spheres about the origin are isoperimetric if and only if log f is convex (the stability condition). We discuss recent progress and open questions for other densities, unequal perimeter and volume densities, and other metrics.

### Quiver Grassmannians of Dynkin type

Giovanni Cerulli-Irelli
Given a finite-dimensional representation M of a Dynkin quiver Q (which is the orientation of a simply-laced Dynkin diagram) we attach to it the variety of its subrepresentations. This variety is strati ed according to the possible dimension vectors of the subresentations of M. Every piece is called a quiver Grassmannian. Those varieties were introduced by Schofield and Crawley Boevey for the study of general representations of quivers. As pointed out by Ringel, they also...

### Interview at CIRM: Pavel Exner

Pavel Exner
Pavel Exner from the Academy of Sciences of the Czech Republic in Prague is president of the European Mathematical Society (2015-2018). He's currently also the scientific director at the Doppler Institute for Mathematical Physics and Applied Mathematics in Prague.

### Computing Hecke operators for cohomology of arithmetic subgroups of $SL_n(Z)$

Mark W. McConnell
We will describe two projects. The first which is joint with Avner Ash and Paul Gunnells, concerns arithmetic subgroups $\Gamma$ of $G = SL_4(Z)$. We compute the cohomology of $\Gamma \setminus G/K$, focusing on the cuspidal degree $H^5$. We compute a range of Hecke operators on this cohomology. We fi Galois representations that appear to be attached to the Hecke eigenclasses, based on the operators we have computed. We have done this for both non-torsion...

### Fourier coefficients of meromorphic Jacobi forms

Sander Zwegers
Fourier coefficients of meromorphic Jacobi forms show up in, for example, the study of mock theta functions, quantum black holes and Kac-Wakimoto characters. In the case of positive index, it was previously shown that they are the holomorphic parts of vector-valued almost harmonic Maass forms. In this talk, we give an alternative characterization of these objects by applying the Maass lowering operator to the completions of the Fourier coefficients. Further, we'll also describe the relation...

### Bayesian inference and mathematical imaging - Part 3: probability and convex optimisation

Marcelo Pereyra
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation...

### A case of the dynamical André-Oort conjecture

Holly Krieger
In the past few years, conjectures have been made and partial results achieved on unlikely intersections in complex dynamics, following the program initiated by Baker-DeMarco. I will explain their dynamical generalization of the famous André-Oort conjecture on CM points in moduli spaces of abelian varieties. Ghioca, Nguyen, Ye, and myself recently proved the first complete case of this conjecture, for pairs of unicritical polynomials, and I will discuss our result and the connection to invariant...

### Big mapping class groups - lecture 3

Danny Calegari
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe...

### Least squares regression Monte Carlo for approximating BSDES and semilinear PDES

Plamen Turkedjiev
In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations. The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the of the continuous...

### Finite type invariants of knots in homology 3-spheres

Delphine Moussard
For null-homologous knots in rational homology 3-spheres, there are two equivariant invariants obtained by universal constructions à la Kontsevich, one due to Kricker and defined as a lift of the Kontsevich integral, and the other constructed by Lescop by means of integrals in configuration spaces. In order to explicit their universality properties and to compare them, we study a theory of finite type invariants of null-homologous knots in rational homology 3-spheres. We give a partial...

### Willmore stability and conformal rigidity of minimal surfaces in $\mathbb{S}^{n}$

Rob Kusner
A minimal surface $M$ in the round sphere $\mathbb{S}^{n}$ is critical for area, as well as for the Willmore bending energy $W=\int\int(1+H^{2})da$. Willmore stability of $M$ is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the $W$-stability of $M$ persists in all higher dimensional spheres if and only if the Laplacian of $M$ has first eigenvalue 2. The square Clifford 2-torus in $\mathbb{S}^{3}$ and the equilateral minimal 2-torus...

### Introduction

Pierre Pudlo
​L'intérêt pour l'intelligence artificielle (IA) s'est considérablement accru ces dernières années et l'IA a été appliquée avec succès à des problèmes de société. Le Big Data, le recueil et l’analyse des données, la statistique se penchent sur l’amélioration de la société de demain. Big Data en santé publique, dans le domaine de la justice pénale, de la sécurité aéroportuaire, des changements climatiques, de la protection des espèces en voie de disparition, etc. ​ ​C'est sur...

### Mathematical and numerical analysis of some fluid structure interaction problems - Lecture 2

Céline Grandmont
Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic structure. The purpose of the present lecture...

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