### Bandits in auctions (& more)

Vianney Perchet
In this talk, I will introduce the classical theory of multi-armed bandits, a field at the junction of statistics, optimization, game theory and machine learning, discuss the possible applications, and highlights the new perspectives and open questions that they propose We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing...

### Individualized rank aggregation using nuclear norm regularization

Sahand Neghaban
In recent years rank aggregation has received significant attention from the machine learning community. The goal of such a problem is to combine the (partially revealed) preferences over objects of a large population into a single, relatively consistent ordering of those objects. However, in many cases, we might not want a single ranking and instead opt for individual rankings. We study a version of the problem known as collaborative ranking. In this problem we assume...

### About mathematical modelling for microbial ecosystems with control and design perspectives

Alain Rapaport
The mathematical model of the chemostat has been extensively studied and extended from the eightees, not only as a mathematical representation of the chemostat device invented in the fifties, but also as a general model of resource/consumer dynamics in microbial ecosystems, such as in marine ecology, food fermentation, waste-water treatment, biotechnology. I will present a survey of some recent and less recent results about extensions of this model, that concern the roles of spatialization, density...

### Capacity expansion games with application to competition in power generation investments

René Aïd
We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the discrete investment stages. Working in a continuous-time diffusion framework, we characterize...

### The weak KPZ universality conjecture. Lecture 3

Milton Jara
The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case. Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures. Lecture 2: The martingale approach and energy solutions of the KPZ equation. Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

### Limits of zeroes of holomorphic differential on stable nodal Riemann surfaces

Samuel Grushevsky
We discuss the current status of the problem of understanding the closures of the strata of curves together with a differential with a prescribed configuration of zeroes, in the Deligne-Mumford moduli space of stable curves.

### Panorama des services numériques de la Pateforme en Ligne pour les Mathématiques

Damien Ferney
Un panorama des Services Numériques disponibles sur la Plateforme en Ligne pour les Mathématiques (PLM). Nous ferons un inventaire des services numériques accessibles à travers le Portail des Mathématiques et décrirons leur utilité dans un cadre collaboratif ou nomade et nous aborderons brièvement leur utilisation et leur configuration.

### Small sumsets in continuous and discrete settings

Anne De Roton
Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain...

### Congruent number problem and BSD conjecture

Shou-Wu Zhang
A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should...

### Central limit theorems for circle packings

Mark Pollicott
Given the Apollonian Circle packing, or something similar, one can consider the distribution of the logarithms of the radii. These can be shown to satisfy a Central Limit Theorem. The method of proof uses iterated function schemes and transfer operators and has applications to other conformal dynamical systems.

### Report on the BMR freeness conjecture

Ivan Marin
I will present arguably the most basic one among the set of conjectures stated in 1998 by Broue, Malle and Rouquier (following early work by Broue and Malle) about the generalized Iwahori-Hecke algebras associated to complex reflection groups. By a combination of several kind of arguments and lots of hand-writen as well as computer-assisted calculations, it seems that a complete proof is now within reach. I will report on recent progress by my PhD student...

### Des nombres aux arbres : comment varier ses exemples d'algorithmes ?

Gilles Dowek
Cet exposé présente un certain nombre d'exemples d'exercices de programmation sur les arbres qui peuvent être effectués dans les premières années d'université. Les programmes étant eux-mêmes des arbres, écrire des programmes qui opèrent sur des arbres permet d'écrire des programmes qui opèrent sur d'autres programmes.

### Spectral decomposition of the principal series category

We will discuss the problem of Langlands duality for the principal series category (alias: D-modules on the semi-infinite flag variety). In particular, we will explain how to relate Whittaker invariants to local systems for the Langlands dual group. This work can be understood as a chiralization of the Arkhipov-Bezrukavnikov theory.

### Statistics on graphs and networks (II)

Ulrike Von Luxburg
Consider a sample of points drawn from some unknown density on $R^d$. Assume the only information we have about the sample are the $k$-nearest neighbor relationships: we know who is among the $k$-nearest neighors of whom, but we do not know any distances between points, nor the point coordinates themselves. We prove that as the sample size goes to infinty, it is possible to reconstruct the underlying density p and the distances of the points...

### A hitchhiker's guide to Khovanov homology - Part I

Paul Turner
There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the...

### Darcy problem and crowd motion modeling

Bertrand Maury
We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models. At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction pressures, and that are set on the...

### Geometric control and sub-Riemannian geodesics - Part I

Ludovic Rifford
This will be an introduction to sub-Riemannian geometry from the point of view of control theory. We will define sub-Riemannian structures and prove the Chow Theorem. We will describe normal and abnormal geodesics and discuss the completeness of the Carnot-Carathéodory distance (Hopf-Rinow Theorem). Several examples will be given (Heisenberg group, Martinet distribution, Grusin plane).

### Spherical splines

Hartmut Prautzsch
The Bézier representation of homogenous polynomials has little and not the usual geometric meaning if we consider the graph of these polynomials over the sphere. However the graph can be seen as a rational surface and has an ordinary rational Bézier representation. As I will show, both Bézier representations are closely related. Further I consider rational spline constructions for spherical surfaces and other closed manifolds with a projective or hyperbolic structure.

Omer Offen

### The sparse cardinal sine decomposition and applications

François Alouges
When solving wave scattering problems with the Boundary Element Method (BEM), one usually faces the problem of storing a dense matrix of huge size which size is proportional to the (square of) the number N of unknowns on the boundary of the scattering object. Several methods, among which the Fast Multipole Method (FMM) or the H-matrices are celebrated, were developed to circumvent this obstruction. In both cases an approximation of the matrix is obtained with...

### Analytic torsion for locally symmetric spaces of finite volume

Werner Müller
This is joint work with Jasmin Matz. The goal is to introduce a regularized version of the analytic torsion for locally symmetric spaces of finite volume and higher rank. Currently we are able to treat quotients of the symmetric space $SL(n,\mathbb{R})/SO(n)$ by congruence subgroups of $SL(n,\mathbb{Z})$. The definition of the analytic torsion is based on the study of the renormalized trace of the corresponding heat operators. The main tool is the Arthur trace formula. I...

### Fractional Poisson process: long-range dependence and applications in ruin theory

Romain Biard
We study a renewal risk model in which the surplus process of the insurance company is modeled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.

### The ordered differential field of transseries

Lou Van Den Dries
The field of Laurent series (with real coefficients, say) has a natural derivation but is too small to be closed under integration and other natural operations such as taking logarithms of positive elements. The field has a natural extension to a field of generalized series, the ordered differential field of transseries, where these defects are remedied in a radical way. I will sketch this field of transseries. Recently it was established (Aschenbrenner, Van der Hoeven,...

### 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.

### Interview at CIRM: Sylvia Serfaty

Sylvia Serfaty
Sylvia Serfaty is a Professor at the Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie Paris 6. Sylvia Serfaty was a Global Distinguished Professor of Mathematics in the Courant Institute of Mathematical Sciences. She has been awarded a Sloan Foundation Research Fellowship and a NSF CAREER award (2003), the 2004 European Mathematical Society Prize, 2007 EURYI (European Young Investigator) award, and has been invited speaker at the International Congress of Mathematicians (2006), Plenary speaker at...

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