Institut de Recherche Mathématique Avancée, UMR 7501
- 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
The 88th Encounter between Mathematicians and Theoretical Physicists took place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS in collaboration with Bogolyubov Laboratory of Moscow State University) on September 8-10, 2011. The theme is : "Discretization in Mathematics and in Physics".
Organizers : Dmitry Millionschikov and Athanase Papadopoulos
The invited speakers include :
Ivan Dynnikov (Moscou)
Vladimir Fock (Strasbourg)
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 are survey talks intended for a general audience.
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
Alexander Veselov - Loughborough University
Yang-Baxter maps and discrete integrability
This talk will be addressed to a non-specialized audience.
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
Abstract: 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
This hierarchy can be also treated as discretization of the
Alexei Penskoi - Univérsité de Moscou, laboratoire Bogolyubov
Laplace transformations and spectral theory of two-dimensional semi-discrete hyperbolic Schroedinger operators
Abstract : 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.
9 septembre 2011
Serguei Nechaev - Université de Paris-Sud
On shock’s statistics in "Tetris" game
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.
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.
Vladimir Fock - Strasbourg
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
Olga Kravchenko - Université Claude Bernard, Lyon
Algebraic operadic structures on links and knots
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
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.
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.
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