# Institut de Recherche Mathématique Avancée, UMR 7501

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Accueil du site > Agenda > Colloques et rencontres > Archives > Agenda 2011 > 88e rencontre entre physiciens théoriciens et mathématiciens : Discrétisation en mathématiques et en physique

## 88e rencontre entre physiciens théoriciens et mathématiciens : Discrétisation en mathématiques et en physique

IRMA, 8-10 september 2011

La 88ème rencontre entre physiciens théoriciens et mathématiciens aura pour thème la discrétisation en mathématiques et en physique.

The 88th Encounter between Mathematicians and Theoretical Physicists will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on September 8-10, 2011. The theme will be : "Discretization in Mathematics and in Physics".

Organizers : Dmitry Millionschikov and Athanase Papadopoulos

The invited speakers include :

Ivan Dynnikov (Moscou)
Piotr Grinevich (Moscou)
Rinat Kashaev (Genève)
Satya Majumdar (LPTMS, Orsay)
Sergei Nechaev (LPTMS, Orsay)
Valentin Ovsienko (Lyon)
Yuri Suris (TU Berlin)
Alexander Veselov (Loughborough)
Jean-Bernard Zuber (Paris 6)

Talks are in English. Some of the talks will be survey talks intended for a general audience.

Graduate students and young mathematicians are welcome. Registration is required (and free of charge), at the following link Hotel booking can be asked for through the registration link.

Dmitry Millionschikov

## 8 septembre 2011

### Rinat Kashaev - Université de Genève

Quantum Teichmüller theory and TQFT

Abstract : Quantum Teichmüller theory leads to specific unitary projective representations of mapping class groups of punctured surfaces, where, among other things, some integrable discrete equations can be realized as mapping class dynamics. These representations arise as representations of bigger algebraic structures called Ptolemy groupoids which can also be thought as parts of certain combinatorial cobordism categories. In this way it is natural to expect that quantum Teichmuller theory is a part of TQFT. I will show that such TQFT does indeed exist. The talk is partially based on a joint work with Joergen Ellegaard Andersen.

### Alexander Veselov - Loughborough University

Yang-Baxter maps and discrete integrability

This talk will be addressed to a non-specialized audience.

Coffee break

### Satya Majumdar - Université de Paris-Sud, Laboratoire de physique théorique et modèles statistiques

Random Convex Hulls and Extreme Value Statistics

Abstract : Convex hull of a set of points in two dimension roughly
describes the shape of the setIn this talkI will
discuss the statistical properties of the convex hull
of a set of N independent planar Brownian paths.
We compute exactly the mean perimeter and the mean area
of this convex hullboth for open and closed paths.
We show that the area and perimeter grows extemely slowly (logarithmically with
increasing population size NThis slow growth is a consequence of
extreme value statistics and has interesting implication in
ecological context in estimating the home range
of a herd of animals with population size N.

### Piotr Grinevich - Université de Moscou et Institut Landau

An integrable at one energy elliptic discretization for the 2-dimensional Schrodinger operator

We show that a special elliptic (5-point) discretization of the 2-dimensional Schrodinger operator in integrable at one energy level. Abstract : Integrability means, that this problem admits a wide class of exact (theta-functional) solutions and infinite-dimensional algebra of symmetries, generates by an analog of Toda hierarchy with 2 discrete spatial variables. This hierarchy can be also treated as discretization of the Novikov-Veselov hierarchy.

coffee break

### Alexei Penskoi - Univérsité de Moscou, laboratoire Bogolyubov

Laplace transformations and spectral theory of two-dimensional semi-discrete hyperbolic Schroedinger operators

Anbtract : In this talk we introduce Laplace transformations of 2D
semi-discrete hyperbolic Schroedinger operators
and show their relation to a semi-discrete 2D Toda
latticeWe develop the algebro-geometric spectral
theory of 2D semi-discrete hyperbolic Schroedinger operators.
Using this spectral theory we investigate spectral
properties of the Laplace transformations of these operators.
This makes it possible to find solutions of the semi-discrete
and discrete 2D Toda lattices in terms of theta-functions.

### 17h00

Boat trip around the old city of Strasbourg (the boat trip is offered to the participants by the Mayor of Strasbourg)

### 19h00

Dinner offered to the participants at the restaurant "petit Bois Vert". The address is: 2 quai de la Bruche (Petite France region in the historica center of Strasbourg)

## 9 septembre 2011

### Serguei Nechaev - Université de Paris-Sud

On shock’s statistics in "Tetris" game

Abstract : We consider a (1 + 1) dimensional ballistic deposition process with next-nearest neighbor interactionwhich belongs to the Kardar-Parisi-Zhang universality classand introduce for this discrete model a variational formulation similar to that for the randomly forced continuous Burgers equationThis allows to identify the
characteristic structures in the bulk of a growing aggregate
(clusters'and crevices''with minimizers and shocks in the
Burgers turbulenceWe find scaling laws that characterize the ballistic deposition patterns in the bulk: the thinning" of the forest of clusters with increasing heightand the size distribution of clustersThe critical exponents are computed using the analogy with the Burgers turbulenceThe relation of the "Tetris" with some integrable lattice models will be discussed.

Coffe break

### Yuri Suris - Technische Universität Berlin

On the Lagrangian structure of integrable quad-equations

Abstract : The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. After having been demonstrated for several particular cases of integrable quad-equations by Bazhanov-Mangazeev- Sergeev and by Lobb-Nijhoff, the flip invariance was given a simple and case-independent proof for all integrable quad-equations in my joint work with Bobenko. This result was also extended to asymmetric quad-equations in my joint work with Boll. Moreover, a new relation for Lagrangians within one elementary quadrilateral was found which seems to be a fundamental building block of the various versions of flip invariance. The talk will be devoted to these results and will provide necessary background informations, as well.

### 14h00

Dimers and integrable systems

Abstract : A.B.Goncharov and R.Kenyon discovered a family of integrable systems enumerated by convex polygons on a plane with vertices in integral pointsThe collection of integrals is given by a partition function of a dimer models on a 2D torus and the phase space has a natural cluster structureWe will explain their construction as well as some examplesfitting into this scheme and taking place on symplectic leaves of loop groupsIn particular we will reinterpret in this way relativistic Toda chainSchwarz-Ovsienko-Tabachnikov discrete integrable system on the space of polygons and certain discretisations of the m-KdV equations

Coffe break

### Olga Kravchenko - Université Claude Bernard, Lyon

Abstract : Algebraic structures, such as Lie, associative or commutative, make their appearances in many areas of mathematics. Recently it turned out that there are a few beautiful variations of these classical algebraic structures. Starting from defining a Leibniz algebra (similar to Lie but without the anti-symmetry condition) J.-L. Loday and his collaborators have discovered several other structures. The right framework for their description comes from algebraic topology and is called Theory of operads. In the talk I will give a brief introduction to operads and algebraic structures. Then I will show some examples of algebraic structures found in the theory of knot and tangle invariants based on my joint work with M. Polyak.

### Valentin Ovsienko - Université Claude Bernard, Lyon

The pentagram map and generalized friezes of Coxeter

This talk will be addressed to a non-specialized audience.

Abstract : The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation)I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram mapIn particularI will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud Richard Schwartz and Serge Tabachnikov.

### 18h00

Reception at the city hall (mairie)offered by the Mayor of Strasbourg.

## 10 septembre 2011

### Ivan Dynnikov - Université de Moscou et Institut Steklov

A discretization of complex analysis

This talk will be accessible to a nonspecialized audience.

Abstract : I will talk about our joint work with S.P.Novikov in which we found a nice discrete analogue of complex derivativesOur difference operators on a triangular lattice mimic the operators $\partial$ and $\overline\partial$ in many key aspectsfactorization of the Laplace operatorthe Liouville principlethe maximum principleTailor expansions and the Cauchy formula.

Coffe break

### Sergey V. Smirnov - Université de Moscou

Integrable semidiscrete Toda lattices

Abstract : Integrable boundary conditions for two-dimensional Toda lattice are known to be described by Cartan matrices of semisimple Lie algebras since the beginning of 1980-es. Various approaches have been used in the continuous case by Mikhailov, Shabat and Yamilov, Adler, Habibullin and Gurel and by many others. One-dimensional discrete Toda chains have been studied by Suris in 1990. The situation in (semi)-discrete case is much more complicate. In both cases no analogs of B and D series are yet known. In purely discrete case the C series was examined by Habibullin in 2006. My talk will be focused on the semidiscrete caseI'll expalain how Lax presentation for the C-series lattice can be obtained using two Lax pairs for the infinite lattice and I'll show how the symmetry approach can be used to classify integrable boundary conditions of a certain type for the semidiscrete Toda lattice.

Dernière mise à jour le 22-07-2011