Schedule
The classroom is 5440 (4th floor).
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 socalled transportentropy
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 logSobolev 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 (BonnetMyers 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 trianglefree graph look like?
2) What is the largest trianglefree subgraph of the random graph $G(n,p)$?
3) When does every rcolouring 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 KhinchineKahane 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 ndimensional Euclidean space.
Short talks
 Anna Aksamit  Robust pricinghedging duality for American options
(presentation)
Theory of MongeKantorovich mass transport has been proven useful in financial applications.
Based on martingale (optimal) transport, modelindependent bounds and their dual representations have been
obtained for the prices of exotic options. We investigate pricinghedging 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 Khintchinetype 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 timechanged random walk
and a timechanged Brownian Motion. We then apply this result to bound the distance between a timechanged compensated scaled Poisson process
and a timechanged Brownian Motion. This allows us to bound the distance between the Moran model with mutation and WrightFisher diffusion with
mutation upon noting that the former may be expressed as a difference of two timechanged Poisson processes and the diffusion part of the latter
may be expressed as a timechanged Brownian Motion. The method is applicable to a much wider class of examples satisfying the StroockVaradhan
theory of diffusion approximation.
 Michał Lemańczyk  The Bernsteinlike concentration inequality for Markov chains.
Firstly we present a method which enables us to obtain Bernsteinlike inequality for socalled onedependent sequences of random variables
in two special cases, namely for socalled twoblockfactors and 1dependent Markov chains. Then we show how using this method we can get
Bersteinlike 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 nonempty 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 KhinchineKahane 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 logSobolev inequalities of BobkovLedoux type for convex and concave functions (and consequently certain weak transportentropy
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.
