### 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...

### Interview au CIRM : David Ruelle

David Ruelle
David Ruelle est professeur honoraire de Physique Théorique à l’Institut des Hautes Études Scientifiques (IHÉS). Il a reçu la médaille Matteucci en 2004, en 2006 le Prix Henri-Poincaré et en 2014 la Médaille Max-Planck pour l'ensemble de ses travaux.

### Linear solvers for reservoir simulation

Pascal Hénon
In this presentation, we will first present the main goals and principles of reservoir simulation. Then we will focus on linear systems that arise in such simulation. The main HPC challenge is to solve those systems efficiently on massively parallel computers. The specificity of those systems is that their convergence is mostly governed by the elliptic part of the equations and the linear solver needs to take advantage of it to be efficient. The reference...

### Theta lifts of tempered representations and Langlands parameters

Wee Teck Gan
In joint work with Hiraku Atobe, we determine the theta lifting of irreducible tempered representations for symplectic-metaplectic–orthogonal and unitary dual pairs in terms of the local Langlands correspondence. The main new tool for proving our result is the recently established local Gross-Prasad conjecture.

### Towards complex and realistic tokamaks geometries in computational plasma physics

Ahmed Ratnani
edge localised modes (ELMs)#MHD simulation (JOREK)#computer aided design (CAD)#isogeometric analysis (IGA)#splines#h-p-k-refinement#k-refinement#flux aligned meshes#ITER tokamak#WEST tokamak#r-refinement#Monge-Kantorovich problem#Monge-Ampère equation#geometric multigrid#two-grid method for nonlinear PDE#equidistribution - adaptative anisotropic mapping

### Orderability and groups of homeomorphisms of the circle

Kathryn Mann
As a counterpart to Deroin's minicourse, we discuss actions of groups on the circle in the C0 setting. Here, many dynamical properties of an action can be encoded by the algebraic data of a left-invariant circular order on the group. I will highlight rigidity and flexibility phenomena among group actions, and discuss new work with C. Rivas relating these to the natural topology on the space of circular orders on a group.

### Minicourse shape spaces and geometric statistics

Xavier Pennec & Alain Trouvé

### Forcing theory for transverse trajectories of surface homeomorphisms - Part 3

Fabio Tal
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...

### Maps between curves and diophantine obstructions

José Felipe Voloch
Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the...

### Schubert calculus and self-dual puzzles

Iva Halacheva
Puzzles are combinatorial objects developed by Knutson and Tao for computing the expansion of the product of two Grassmannian Schubert classes. I will describe how selfdual puzzles give the restriction of a Grassmannian Schubert class to the symplectic Grassmannian in equivariant cohomology. The proof uses the machinery of quantum integrable systems. Time permitting, I will also discuss some ideas about how to interpret and generalize this result using Lagrangian correspondences and Maulik-Okounkov stable classes. This...

### Approximation and calibration of laws of solutions to stochastic differential equations

In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations. This new...

### Understanding quadratic forms on lattices through generalised theta series

Lynne Walling
Siegel introduced generalised theta series to study representation numbers of quadratic forms. Given an integral lattice $L$ with quadratic form $q$, Siegel’s degree $n$ theta series attached to $L$ has a Fourier expansion supported on $n$-dimensional lattices, with Fourier coefficients that tells us how many times $L$ represents any given $n$-dimensional lattice. Siegel proved that this theta series is a type of automorphic form. In this talk we explore how the theory of automorphic forms,...

### On Schmidt's subspace theorem

Jan-Hendrik Evertse
Last year, I published together with Roberto Ferretti a new version of the quantitative subspace theorem, giving a better upper bound for the number of subspaces containing the solutions of the system of inequalities involved. In my lecture, I would like to discuss this improvement, and go into some aspects of its proof.

### Multiple ergodic theorems: old and new - Lecture 1

Bryna Kra
The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

### On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications

Francisco José Silva Álvarez
In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity of the solution, conserves the mass and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. This convergence result is proved by assuming only that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, under these assumptions,...

### Formal conjugacy growth and hyperbolicity

Laura Ciobanu
Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. In this talk I will present the proof (joint with Hermiller, Holt and Rees) that the conjugacy growth series of a virtually cyclic group is rational, and then also confirm the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental (joint with Antolín)....

### Markov Chain Monte Carlo Methods - Part 1

Christian P. Robert
In this short course, we recall the basics of Markov chain Monte Carlo (Gibbs & Metropolis sampelrs) along with the most recent developments like Hamiltonian Monte Carlo, Rao-Blackwellisation, divide & conquer strategies, pseudo-marginal and other noisy versions. We also cover the specific approximate method of ABC that is currently used in many fields to handle complex models in manageable conditions, from the original motivation in population genetics to the several reinterpretations of the approach found...

### Continuous and discrete uncertainty principles

Bruno Torrésani
Uncertainty principles go back to the early years of quantum mechanics. Originally introduced to describe the impossibility for a function to be sharply localized in both the direct and Fourier spaces, localization being measured by variance, it has been generalized to many other situations, including different representation spaces and different localization measures. In this talk we first review classical results on variance uncertainty inequalities (in particular Heisenberg, Robertson and Breitenberger inequalities). We then focus on...

### Arithmetic $D$-modules and existence of crystalline companion

Tomoyuki Abe
We will show that there exists a correspondence between smooth $l$-adic sheaves and overconvergent $F$-isocrystals over a curve preserving the Frobenius eigenvalues. Moreover, we show the existence of $l$-adic companions associated to overconvergent $F$-isocrystals for smooth varieties. Some part of the work is done jointly with Esnault.

### Distributions of Frobenius of elliptic curves #4

Nathan Jones
In the 1970s, S. Lang and H. Trotter developed a probabilistic model which led them to their conjectures on distributional aspects of Frobenius in $GL_2$-extensions. These conjectures, which are still open, have been a significant source of stimulation for modern research in arithmetic geometry. The present lectures will provide a detailed exposition of the Lang-Trotter conjectures, as well as a partial survey of some known results. Various questions in number theory may be viewed in...

### 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...

### Mahler's method in several variables

Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler’s method provides one of the few known instances where it essentially...

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