CNRS-IMPAN Mathematics Summer Institute, Cracow 2 - 9 July, 2017



In collaboration with the Institute of Mathematics, Jagiellonian University, Institute of Mathematics, Polish Academy of Sciences, CNRS, Imperial College London, IMP Universite Paul Sabatier Toulouse, and ANR project STAB.

Supported by Warsaw Center of Mathematics and Computer Sciences.

Organizers: Dominique Bakry (Toulouse), Szymon Peszat (Cracow), and Bogusław Zegarliński (London).

The meeting will review recent results in the area of analysis/stochastics. Besides a number of presentation by international participants, the meeting will include two mini-courses (suitable for PhD students and young researchers). It is probable that we will be able to cover the accommodation of some number of participants (with the strong emphasis of young people and researchers without financial support of their organizations). In order to register, send an e-mail to: Szymon Peszat on e.mail adress napeszat@cyf-kr.edu.pl

The venue of the workshop is: Room 0094, the main building of the Faculty of Mathematics and Computer Sciences, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków.

Previous and related activities: Minicourses:
  • Sergey Bobkov (Minneapolis): A second order concentration of measure on the sphere, and its application to randomized central limit theorems.
    • Abstract: There will be discussed deviation inequalities for smooth functions on the sphere with improved rates relying upon the second partial derivatives (Hessians). Such results are illustrated in the problem of rates of normal approximation for weighted sums of dependent random variables under proper second order correlation-type conditions.
  • Persi Diaconis (Stanford): The mathematics of shuffling cards.
    • Lecture 1 Introduction. These lectures will prove theorems about real world methods of shuffling cards (e.g., it takes about 7 riffle shuffles to mix up 52 cards). The first lecture sets up the problem and proves the 7 shuffles theorem.
    • Lecture 2 Adding numbers. When numbers are added in the usual way, carries occur along the way. It turns out that the carries form a Markov chain with an "amazing" transition matrix. Strange to say, these problems are closely related to the mathematics of riffle shuffling.
    • Lecture 3 Hyperplane walks. There is an elegant family of random walks on the chambers of a hyperplane arrangement. This includes various shuffling schemes. All these walks are explicitly diagonalizable and have "closed form" stationary distributions.
    • Lecture 4 shuffling cards and Hopf algebras. Hopf algebras are algebraic "gadgets" used by topologists. In many examples, the "Hopf square map" has a simple probabilistic interpretation. The free associative algebra leads to riffle shuffles, symmetric functions leads to a rock breaking model of Kolmogorov. The hard work done by combinatorialists and others gives explicit forms of the eigenvectors.
    • Lecture 5 Overhand shuffling. This commonly used method of mixing needs different sets of tools for analysis, coupling, and comparison theory will be introduced and do a pretty good job.
    • Lecture 6 Smooshing. The "wash of smoosh" shuffle is widely used in poker tournaments and in Monte Carlo. It involves sliding the cards around on the table to mix them. I will introduce a fluid mechanics model to study this and a novel coupling technique to get quantitative results.


Registered Participants: Dominique Bakry (Toulouse), Oliwier Biernacki (Wrocław), Sergey Bobkov (Minneapolis), Dariusz Buraczewski (Wrocław), Adam Burchardt (Poznań), Persi Diaconis (Stanford), Jose Carillo (London), Jose A. Canizo (Granada), Klaudiusz Czudek (Gdańsk), Gracjan Góral (Wrocław), Tomasz Kosmala (London), Kajetan Jastrzębski (Wrocław), Zbigniew Jurek (Wrocław), Xu Lihu (Macau), Paweł Lorek (Wrocław), Piotr Markowski (Wrocław), Laurent Miclo (Toulouse), Wojtek Słomczyński (Cracow), Roman Srzednicki (Cracow), Miroslav Strupl (Prague), Tomasz Szarek (Gdańsk), Maciej Szczeciński (Zielona Góra), Jacek Wesołowski (Warsaw), Robert Wolstenholme (Prague), Filip Zagórski (Wrocław), Frantisek Zak (Prague), Andrzej Zuk (Paris).

Talks:

Dariusz Buraczewski - Large deviations for random walks in random environment.

Jose A. Canizo - Entropy production inequalities for the linear Boltzmann equatiion.

Tomasz Kosmala - Variational solutions to SPDEs driven by cylindrical Levy noise. Abstract: We prove the existence and uniqueness of solution to an infinite dimensional evolution equation driven by a cylindrical Levy process. It is assumed that the coefficients in the equation are monotone and coercive. The solution is constructed as a limit of the Galerkin approximation by projecting the equation onto n-dimensional subspaces, which enables us to use results from finite dimension.

Zbigniew Jurek - (Some) Ising models and the selfdecomposability property. Abstract: Selfdecomposable distributions ( called also as the class L distributions) appear as an extention of CLT. It is a quite big class that includes stable measures, t-Student, Snedecor F, gamma, etc. We will show how some of selfdecomposable distributions can be related to the Ising model for ferromagnetism with the external (magnetisation) field.

Laurent Miclo - On Markov intertwinings. Abstract: After recalling the intertwining relation of the Brownian motion with the Bessel 3 process due to Pitman (1975) and the use of a corresponding technique to deal with the convergence to equilibrium for the top-to-random card shuffle by Aldous and Diaconis (1986), we will extend this procedure to elliptic diffusions on manifolds via stochastic modifications of mean curvature flows.

Tomasz Szarek - Random function systems on the circle. Abstract: The talk will be devoted to ergodicity of random function systems on the circle. We show also that such systems, under very weak conditions, satisfy the law of large numbers and central limit theorem.

Lihu Xu - Convergence rate of stable law. Abstract: By Stein's method, we prove a general inequality of stable law. As an application, we give the convergence rate for an example which appears in many textbooks.

Andrzej Zuk - Random walks on ultra discrete limits.

Program:

Sunday 2 July Arrival Day

Monday 3 July

10.00 - 10.30 T&C
10.30 - 12.00 Persi Diaconis (Mini-course 1)

Lunch

13.30 - 15.00 Sergey Bobkov (Mini-course 1)
15.00 - 15.30 T&C
15.30 - 16.15 Jose A. Canizo (Entropy production inequalities for the linear Boltzmann equatiion)
16.20 - 16.50 Tomasz Kosmala (Variational solutions to SPDEs driven by cylindrical Levy noise)

Tuesday 4 July

10.00 - 10.30 T&C
10.30 - 12.00 Sergey Bobkov (Mini-course 2)

Lunch

13.30 - 15.00 Persi Diaconis (Mini-course 2)
15.00 - 15.30 T&C
15.30 - 16.15 Laurent Miclo (On Markov intertwinings)
16.20 - 17.05 Zbigniew Jurek ((Some) Ising models and the selfdecomposability property)

Wednesday 5 July

10.00 - 10.30 T&C
10.30 - 12.00 Sergey Bobkov (Mini-course 3)

Thursday 6 July

10.00 - 10.30 T&C
10.30 - 12.00 Persi Diaconis (Mini-course 3)

Lunch

13.30 - 14.15 Tomasz Szarek (Random function systems on the circle)
14.15 - 14.30 T&C
14.30 - 15.15 Lihu Xu (Convergence rate of stable law)
15.30 - 16.15 Dariusz Buraczewski (Large deviations for random walks in random environment)

Friday 7 July

10.00 - 10.30 T&C
10.30 - 12.00 Persi Diaconis (Mini-course 4)

Lunch

13.30 - 14.15 Andrzej Zuk (Random walks on ultra discrete limits)
14.15 - 14.30 T&C