### 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.

### Solitons vs collapses

Evgenii Kuznetsov
This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves,...

### Brauer-Siegel theorem and analogues for varieties over global fields

Marc Hindry
The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant. This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1. Even if some of the fundamental problems like...

### Almost sure scattering for the energy-critical Schrödinger equation in 4D with radial data

Monica Visan
Inspired by a recent result of Dodson-Luhrmann-Mendelson, who proved almost sure scattering for the energy-critical wave equation with radial data in four dimensions, we establish the analogous result for the Schrödinger equation. This is joint work with R. Killip and J. Murphy.

### Learning on the symmetric group

Jean-Philippe Vert
Many data can be represented as rankings or permutations, raising the question of developing machine learning models on the symmetric group. When the number of items in the permutations gets large, manipulating permutations can quickly become computationally intractable. I will discuss two computationally efficient embeddings of the symmetric groups in Euclidean spaces leading to fast machine learning algorithms, and illustrate their relevance on biological applications and image classification.

### Numerical methods for mean field games - Lecture 3: Variational MFG and related algorithms for solving the discrete system of nonlinear equations

Yves Achdou
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and...

### Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes

Yves Achdou
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and...

### Splitting algorithm for nested events

Ludovic Goudenège
Consider a problem of Markovian trajectories of particles for which you are trying to estimate the probability of a event. Under the assumption that you can represent this event as the last event of a nested sequence of events, it is possible to design a splitting algorithm to estimate the probability of the last event in an efficient way. Moreover you can obtain a sequence of trajectories which realize this particular event, giving access to...

### Project evaluation under uncertainty

Jorge P. Zubelli
Industrial strategic decisions have evolved tremendously in the last decades towards a higher degree of quantitative analysis. Such decisions require taking into account a large number of uncertain variables and volatile scenarios, much like financial market investments. Furthermore, they can be evaluated by comparing to portfolios of investments in financial assets such as in stocks, derivatives and commodity futures. This revolution led to the development of a new field of managerial science known as Real...

### Mean field type control with congestion

Mathieu Laurière
The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively...

### Interview at CIRM: Martin Hairer

Martin Hairer
Martin Hairer KBE FRS (born 14 November 1975 in Geneva, Switzerland) is an Austrian mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. As of 2017 he is Regius Professor of Mathematics at the University of Warwick, having previously held a position at the Courant Institute of New York University. He was awarded the Fields Medal in 2014, one of the highest honours a mathematician can achieve.

### Computing Sato-Tate statistics

Andrew Sutherland
Survey of methods for computing zeta functions of low genus curves, including generic group algorithms, p-adic cohomology, CRT-based methods (Schoof-Pila), and recent average polynomial-time algorithms. Sato-Tate - Abelian surfaces - Abelian threefolds - hyperelliptic curves

### Group structures of elliptic curves #1

Igor Shparlinski
We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like. This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying...

### Group structures of elliptic curves #2

Igor Shparlinski
We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like. This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying...

### Distributions of Frobenius of elliptic curves #6

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...

### Distributions of Frobenius of elliptic curves #2

Chantal David
In all the following, let an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication. For every prime $\ell$, let $E[\ell]= E[\ell](\overline{\mathbb{Q}})$ be the group of $\ell$-torsion points of $E$, and let $K_\ell$ be the field extension obtained from $\mathbb{Q}$ by adding the coordinates of the $\ell$-torsion points of $E$. This is a Galois extension of$\mathbb{Q}$ , andGal$(K_\ell/\mathbb{Q})\subseteq GL_2(\mathbb{Z}/\ell\mathbb{Z})$. Using the Chebotarev density theorem for the extensions $K_\ell/\mathbb{Q}$ associated to a given curve $E$,...

### The generalized Sato-Tate conjecture

Francesc Fité
This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the second talk, we present the Sato-Tate axiomatic, which leads...

### Sato-Tate axioms

Francesc Fité
This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the second talk, we present the Sato-Tate axiomatic, which leads...

### The Galois type of an Abelian surface

Francesc Fité
This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the second talk, we present the Sato-Tate axiomatic, which leads...

### Some news on bilinear decomposition of the Möbius function

Olivier Ramaré
This talk presents some news on bilinear decompositions of the Möbius function. In particular, we will exhibit a family of such decompositions inherited from Motohashi's proof of the Hoheisel Theorem that leads to $\sum_{n\leq X,(n,q)=1) }^{} \mu (n)e(na/q)\ll X\sqrt{q}/\varphi (q)$ for $q \leq X^{1/5}$ and any $a$ prime to $q$.

### Arithmetic of algebraic points on varieties over function fields - Part 3

Carlo Gasbarri
We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields.

### Integrable probability - Lecture 2

Ivan Corwin
A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent...

### Two-weight inequalities meet $R$-boundedness

Tuomas P. Hytönen
One of my recent main interests has been the characterization of boundedness of (integral) operators between two $L^p$ spaces equipped with two different measures. Some recent developments have indicated a need of "Banach spaces and their applications" also in this area of Classical Analysis. For instance, while the theory of two-weight $L^2$ inequalities is already rich enough to deal with a number of singular operators (like the Hilbert transform), the $L^p$ theory has been essentially...

### The story of Kalton's last unpublished paper

Jesús M.F. Castillo
I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not. In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then $\mathbb{Z}^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space, and one gets from here...

### Multi-norms and Banach lattices

H. Garth Dales
I shall discuss the theory of multi-norms. This has connections with norms on tensor products and with absolutely summing operators. There are many examples, some of which will be mentioned. In particular we shall describe multi-norms based on Banach lattices, define multi-bounded operators, and explain their connections with regular operators on lattices. We have new results on the equivalences of multi-norms. The theory of decompositions of Banach lattices with respect to the canonical 'Banach-lattice multi-norm'...

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