June 23

11:00 Frederic Chapoton, "Combinatorics of shrubs"

We will explain the new combinatorial notion of *shrub*, which has appeared in the study of an operad related to shuffles. Shrubs can be seen as some kind of generalized "Hasse diagrams" or as some sort of generalized rooted trees. We will first explain the combinatorial aspects, then deal with more algebraic things, if time permits.

12:00 A.B.Sossinsky, The kinematics of mechanical linkages with singular moduli spaces

The talk will begin with a brief overview of some recent results about the moduli spaces of mechanical linkages (hinge mechanisms in another terminology), including universality theorems due to Mnev, Thurston, King, and others. We will then concentrate on the particular case of the so-called pentagons and hexagons and describe their classification, following the work of D.Zvonkine, Steiner and Curtis, Kapovich and Millman, and the author. Here the main point is the construction of a Vassiliev-type invariant on the discriminant of the parameter space of pentagons; this invariant classifies the moduli spaces of the latter up to real algebraic isomorphism.

Further, the talk will address the case of singular (nongeneric) moduli spaces (which are real algebraic varieties with singularities) in order to describe the kinematics of mechanical linkages in the case when the moduli spaces are not smooth manifolds. Thus we define tangent vectors, tangent planes (which are not linear spaces and are topologically nontrivial!), and smooth vector fields at singular points and discuss smooth trajectories of mechanical linkages in the nonsmooth situation.

In conclusion, we will speculate about the dynamics of mechanical linkages in the nonsmooth situation and state a series of concrete problems whose ultimate goal is the construction of the differential and integral calculus on real algebraic manifolds with singularities.

13:00 Evgeny Smirnov, Schubert decomposition for double Grassmannians and a partial order on involutive permutations

Classical Schubert calculus deals with orbits of a Borel subgroup B in GL(V) acting on a Grassmann variety Gr(k,V) of k-planes in a finite-dimensional vector space V. These orbits (Schubert cells) and their closures (Schubert varieties) are very well studied both from the combinatorial and the geometric points of view.

One can go one step farther, considering the direct product of two Grassmannians Gr(k,V)x Gr(l,V) and the Borel subgroup B in GL(V) acting diagonally on this variety. In this case, the number of orbits still remains finite, but their combinatorics and geometry of their closures become much more involved.It would be challenging to extend the whole body of the Schubert calculus to this situation.

In my talk I will focus on the combinatorial part of this problem. I will explain how to index the B-orbit closures in Gr(k,V)x Gr(l,V) by means of rook placements in Young diagrams. This construction provides a partial order on involutive permutations, that also appeared earlier in some other situations, in particular, in the papers by A.Melnikov on nilpotent adjoint B-orbits and in L.Renner's studies of affine algebraic monoids.

14:30 Lunch

16:00 Discussions

June 24

10:00 Gleb Oshanin, Random patterns generated by random permutations of natural numbers

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part (joint work with R.Voituriez), we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are prescribed by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories - permutation "peaks" and "trough". In the second part (joint work with F.Hivert, S.Nechaev and O.Vasilyev), we focus on some statistical properties of surfaces obtained by randomly placing natural numbers 1, 2, 3,..., L on sites of a Id or 2d lattices containing L sites. We calculate the distribution function of the number of local "peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites and discuss surprising collective behavior emerging in this model.

11:00 Jean-Christophe Novelli, New applications of combinatorial Hopf algebras

We will discuss recent applications of classical combinatorial Hopf algebras (noncommutative symmetric functions, quasi-symmetric functions, free quasi-symmetric functions, word quasi-symmetric functions, ...) to various problems outside the world of combinatorics.

Coffer break

12:10 Jean-Yves Thibon, Tree expanded series in some combinatorial Hopf algebras

Tree expanded series occur as formal solutions of nonlinear functional equations, such as Schwinger-Dyson equations in QFT. However, the tree expansions of the solutions of simple differential equations already lead to interesting combinatorics when lifted to certain Hopf algebras. For example, the trivial fact that the geometric series $x(t)=1/(1-t)$ satisfies $x'(t)=x^2$ has nontrivial consequences, such as the Bj?rner-Wachs q-hook-length formulas for decreasing binary trees. The algebraic mechanism underlying such identities can then be traced back to some dendriform structure, which leads to various generalizations involving two parameters.

13:10 Gleb Koshevoy, Tropical Plucker functions, their bases and MV-polytopes

Tropical Plucker functions is a class of functions that obey tropical analogs of classical Plucker relations on minors of a matrix. We construct a basis for the set of tropical Plucker functions defined on integer points of truncated box $\mathbf B_m^{m'}$, that is a subset $B\subset \mathbf B_m^{m'}$, such that the restriction map $TP(\mathbf B_m^{m'})\to \mathbb R^B$$, bijective. Also we characterize, in terms of the restriction to the basis, the classes of submodular, so- called skew-submodular, and discrete concave functions in T P, and present a bijection between MV-polytopes and submodular TP-functions on a Boolean cube. This is a joint work with V. Danilov and A.Karzanov

14:30 Lunch

16:00 Discussions

June 25

Free day

June 26

10:00 Frederic Patras, One-particle irreducibility with initial correlations

In quantum field theory (QFT), the vacuum expectation value of a normal product of creation and annihilation operators is always zero. This simple property paves the way to the classical treatment of perturbative QFT by means of Feynman diagrams. This is no longer the case in the presence of initial correlations, that is if the vacuum is replaced by a general state. As a consequence, the combinatorics of correlated systems such as the ones occuring in many-body physics is more complex than that of quantum field theory and the general theory has made very slow progress. Similar observations hold in statistical physics or quantum probability for the perturbation series arising from the study of non Gaussian measures. In this talk, an analysis of the Hopf algebraic aspects of quantum field theory is used to derive the structure of Green functions in terms of connected and one-particle irreducible Greeen functions for perturbative QFT in the presence of initial correlations. Joint work with Ch. Brouder and A. Frabetti.

11:00 Vladimir Danilov, Wirings, weakly separated set-systems, and the Leclerc-Zelevinsky conjectures

In relations to study of bases of tropical Plucker functions, we introduce and study a class of wirings diagrams in a disc, which generalizes the class of wirings diagram when two diffrent wires have no more one common point. Bases of such diagramms we construct a class of plane acyclic digraphs. We will show that the vertices of such a digraph form a weakly separated set-system, and any pair of strongly separated sets in such a system is connected by path. Using these results we affirmatively answer all conjectures by Leclerc and Zelevinsky (1998) on weakly-separated sets. This is joint work with A.Karzanov and G.Koshevoy

Coffer Break

12:10 Yu.Burman, A refinement of Hurwitz numbers and one-faced graph embeddings

Hurwitz numbers H_{g,l} enumerate factorizations of a permutation of the cyclic type l = (l_1, ..., l_s) into a product of m = 2g-2+s transpositions. Alternatively, they count covers of the projective line by a genus g curve with ramification pattern (l_1, ..., l_s) over infinity and m simple ramification points. We will try, instead of just counting the factorizations, to retain information about the transpositions used. This makes Hurwitz number a polynomial of the set of doubly indexed variables w_{ij}. For the simplest case of the cyclic ramification (s=1) this polynomial admits a closed formula. The result obtained can be used to count the homotopy classes of embeddings of any given graph into a surface such that the complement of the graph is homeomorphic to a disk.

The talk is based on the results obtained jointly with D.Zvonkine.

13:10 Denis Mironov, Seperation index of root systems

This is a joint work with V. Zhgoon. Separation index of a root system (and more generally of reflection group) was introduced in 2006 by V.L. Popov as combinatorial parameter of Weyl Chambers (or translations of fundamental cone). This talk is about relation of separation index to primitive weight tuples and tensor decompositions of representations. In second part I will present some precise computations and upper bounds on separation index.

14:30 Lunch

16:00 Discussions

June 27

10:30 Alexander Karzanov, Plucker environments and generalized tilings of subsets of 2n-gones

We consider the class of bases B of tropical Pl?ucker functions on a truncated Boolean n-cube such that B can be obtained by a series of flips from the basis formed by the intervals of the ordered set of n elements. We show that these bases are representable by generalizing rhombus tilings on a zonogon. This is a joint work with V.Danilov and G.Koshevoy

11:30 Yurii Neretin, Title "Point configurations on 3-dimensional sphere and outer automorphisms of free groups"

We consider the space of point configurations on the 3-dimensional sphere upto rotations. We construct the action of the group of outer automorphisms of free group on this space. We also construct action of a braid group on the space of closed polygonal curves in $\S^3$ with given lengths of edges.