## 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.

## 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.

#### 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.