Abstracts

Long courses

Each of the following lecturers will deliver four 90 minute lectures:

• Nathael Gozlan - Optimal transport and concentration of measure

These lectures will be devoted to applications of optimal transport theory to functional inequalities and concentration of measure phenomena.
The first lecture will recall basic tools from optimal transport theory (in particular Kantorovich duality theorem and Brenier's theorem about the existence of an optimal transport map).
The second lecture will introduce the notion of concentration of measure and recall some of its classical applications.
The third and fourth lectures will present the so-called transport-entropy inequalities, a class of functional inequalities introduced by Marton and Talagrand in the nineties. We will see how these inequalities can be used to characterize various types of concentration of measure phenomena.

• Cyril Roberto - Ricci curvature and functional inequalities in discrete spaces

We will first introduce different classical functional inequalities, on graphs: namely the Poincaré Inequality, the log-Sobolev Inequality together with some of its modifications, and a transport inequality. We may explain the relations between them and give their consequence for example in the speed of convergence to equilibrium. Then, after a brief presentation of at least 4 equivalent definitions of the usual Ricci Curvature in continuous spaces, we will review some of their generalizations, in discrete spaces. For each definition we may try to give their implications in terms of functional inequalities, and their geometric aspects, if any (concentration/isoperimetry, bounds on the diameter of the graph (Bonnet-Myers type theorem) etc.). Some explicit graphs and transition kernel (the two point space, the hypercube, the discrete line etc.) will serve as a guideline.

• Wojciech Samotij - The method of hypergraph containers

We shall give an introduction to a recently developed technique in probabilistic combinatorics and asymptotic enumeration called the method of hypergraph containers. This general technique has been successfully applied in a wide range of settings. To give just a few examples, it has been used to study the following questions:

1) What does a typical triangle-free graph look like?
2) What is the largest triangle-free subgraph of the random graph $G(n,p)$?
3) When does every r-colouring of the edges of $G(n,p)$ contain a monochromatic copy of a triangle?

The solutions to these problems exploit the same universal phenomenon, which the container method describes in a precise quantitative sense: A very large class of combinatorial structures satisfying a family of "local" constraints exhibits a certain kind of "clustering" phenomenon.

Educational lectures

• Octavio Arizmendi - Asymptotic Freeness: A connection between free groups and random matrices (presentation)

In this talk we will explain the phenomenon of asymptotic freeness. For this we will start with basics on non commutative probability, free independence and free convolution. After this we will explain the main result of Voiculescu about the behavior of Random Matrices with certain symmetries: Asymptotic Freeness. We will finish with more specific examples.

• Krzysztof Oleszkiewicz - Probabilistic inequalities on the discrete cube (presentation)

We will discuss two connections between probability and harmonic analysis on the discrete cube: the Khinchine-Kahane inequality and the FKN theorem.

• Joscha Prochno - Geometry of random polytopes (lecture notes)

In this lecture we will take a look at the expected mean width of a random polytope that is generated by N random points drawn uniformly at random from an isotropic convex body K in n-dimensional Euclidean space.

Short talks

• Anna Aksamit - Robust pricing--hedging duality for American options (presentation)

Theory of Monge-Kantorovich mass transport has been proven useful in financial applications. Based on martingale (optimal) transport, model-independent bounds and their dual representations have been obtained for the prices of exotic options. We investigate pricing--hedging duality for a more general class of financial instruments, namely for American options. Based on joint work with S. Deng, J. Obłój and X. Tan.

• Giovanni Conforti - A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost (presentation)

Schrödinger bridges provide with a natural probabilistic counterpart of some key notions in optimal transport and lift from the point to the measure setting the concept of brownian bridge. In particular, they are the mathematical object suitable to describe a generalisation of the "lazy gas experiment". Such thought experiment is contained in the original formulation of the Schrödinger problem and can be described as a "hot gas experiment". We prove that Schrödinger bridges are solutions to a second order equation in the Riemannian structure of optimal transport, where the acceleration term is given by the gradient of the Fisher information and, studying the evolution of the marginal entropy, we obtain a quantitative description of the hot gas experiment. As a by product of this analysis, we derive a new functional inequality generalising Talagrand's transportation inequality by replacing the transportation cost with the entropic transportation cost. Some consequences of this inequality are also discussed.

• Aleksandros Eskenazis - Gaussian mixtures and geometric inequalities

A random variable is called a (centered) Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. We will explain how Gaussian mixtures appear in various probabilistic extremization problems and provide extensions of known geometric inequalities about the Gaussian measure. Time permitting, we may also discuss related recent work on sharp Khintchine-type inequalities for random vectors uniformly distributed on the unit ball of $\ell_q^n$. The talk is based on joint work with P. Nayar and T. Tkocz.

• Uri Grupel - Indistinguishable sceneries on the Boolean hypercube (presentation)

Let $Q = \{-1, 1\}^n$ be the n dimensional hypercube. Let $f$ be a labeling of $Q$ by $\{-1, 1\}$. Let $S_n$ be a random walker on $Q$. Can you find the labeling $f$ (up to an automorphism) by knowing $f(S_1)$, $f(S_2)$,...? This question was studied for other graphs such as circles and the integers. We show that for the hypercube it is impossible. To do so, we introduce the class of locally biased functions, and find many non isomorphic functions of this class that induce the same distribution of the scenery $f(S_1)$, $f(S_2)$,... Based on joint work with R. Gross.

• Mikołaj Kasprzak - Diffusion approximations via Stein's method and time changes (presentation)

We extend the ideas of (Barbour, 1990) and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a time-changed compensated scaled Poisson process and a time-changed Brownian Motion. This allows us to bound the distance between the Moran model with mutation and Wright-Fisher diffusion with mutation upon noting that the former may be expressed as a difference of two time-changed Poisson processes and the diffusion part of the latter may be expressed as a time-changed Brownian Motion. The method is applicable to a much wider class of examples satisfying the Stroock-Varadhan theory of diffusion approximation.

• Michał Lemańczyk - The Bernstein-like concentration inequality for Markov chains.

Firstly we present a method which enables us to obtain Bernstein-like inequality for so-called one-dependent sequences of random variables in two special cases, namely for so-called two-block-factors and 1-dependent Markov chains. Then we show how using this method we can get Berstein-like concentration inequality for general Markov chains exploiting a technique called splitting.

• Rafał Meller - Two sided moment estimates for random chaoses (presentation)

Let $X_1,...,X_n$ be the random variables such that there exists a constant $C>1$ satisfying $||X_i||_{2p} \leq C ||X_i||_p$ for every $p \geq 1$. We define random chaos $S=\sum a_{i_1,...,i_d} X_{i_1}...X_{i_d}$. We will show two sided deterministic bounds on $||S||_p$, with constant depending only on $C$ and $d$ in two cases:
1) $X_1,...,X_n$ are a.s. nonnegative, and $a_{i_1,...,i_d}\geq 0$.
2) $X_1,...,X_n$ are symmetric, $d=2$.

• Somabha Mukherjee - Poisson Limit of the number of monochromatic cliques in a uniformly coloured graph (presentation)

We will show that convergence of the first two moments of the number of monochromatic cliques in a uniformly coloured random graph is enough to guarantee an asymptotic Poisson distribution of the number of monochromatic cliques. Usual Poisson convergence results are hard to apply to this problem, due to the absence of joint independence of some particular collection of graphs, even in presence of pairwise independence of this collection. This is a joint work by me and Bhaswar Bhattacharya in the Department of Statistics, University of Pennsylvania.

• Marta Strzelecka - Comparison of weak and strong moments for vectors with independent coordinates (presentation)

We will try to tackle the following problem: "Characterize random vectors for which weak and strong moments are comparable".
As it turns out, such a comparison holds for vectors with independent coordinates with $\alpha$-regular growth of moments:

Theorem Let $X_1,\ldots,X_n$ be independent mean zero random variables with finite moments such that $\|X_i\|_{2p} \le \alpha \|X_i\|_p$ for every $p\ge 2$ and $i=1,\ldots,n$, where $\alpha$ is a finite positive constant.
Then for every $p\ge 1$ and every non-empty set $T\subset \mathbb{R}^n$ we have
$({\mathbb E} \sup_{t\in T} |\sum_{i=1}^n t_iX_i |^p)^{1/p} \le C_\alpha [ {\mathbb E} \sup_{t\in T} |\sum_{i=1}^n t_iX_i| + \sup_{t\in T} ({\mathbb E} |\sum_{i=1}^n t_i X_i |^p )^{1/p} ]$,
where $C_\alpha$ is a constant which depends only on $\alpha$.

Moreover, in the case of i.i.d. coordinates the comparison of weak and strong moments implies $\alpha$-regular growth of moments (with a constant $\alpha$ depending on $C$ only), so the problem posed in the beginning is solved for vectors with i.i.d. coordinates. We will also discuss the consequences, such as a deviation inequality for $\sup_{t\in T}|\sum_{i=1}^n t_iX_i |$ and a Khinchine-Kahane type inequality. The talk will be based on joint work with Rafał Latała

• Michał Strzelecki - On the convex Poincaré inequality (and weak transportation inequalities) (presentation)

We will prove that if a probability measure on $\mathbb{R}^n$ satisfies the Poincaré inequality for convex functions, then it also satisfies modified log-Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions (and consequently certain weak transport-entropy inequalities). This generalizes results by Gozlan et al. and Feldheim et al., concerning probability measures on the real line. Based on joint work with Radosław Adamczak.

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