## Geometry, Combinatorics, and Integrable Systems## Spring 2017Time: Thursdays 3-4pmLocation: MA 317 |
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(Gekhtman): I will discuss a Hamiltonian formalism for cluster mutations using canonical (Darboux) coordinates and piecewise-Hamiltonian flows with Euler dilogarithm playing the role of the Hamiltonian. The Rogers dilogarithm then appears naturally in the dual Lagrangian picture. I will show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from Hamiltonian/Lagrangian point of view. (Based on the joint project with T. Nakanishi and D. Rupel.)

(He): The rational cohomology of real and oriented Grassmannians was given in terms of Pontryagin classes and Euler classes by Leray (1949), Borel (1953) and Takeuchi (1962). In this talk we will consider certain torus actions on these Grassmannians and their rational equivariant cohomology using localization tools.

(Anderson): The essential set of a permutation, defined via its Rothe diagram by Fulton in 1992, gives a minimal list of rank conditions cutting out the corresponding Schubert variety in the flag manifold. I will describe an analogous notion for signed permutations, giving minimal conditions for Schubert varieties in flag varieties for other classical groups. This is related to a poset-theoretic construction of Reiner, Woo, and Yong, and thus gives a diagrammatic method for computing the latter.

(Lando): The talk will present a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and combinatorics of symmetric groups and their representations will be explained. Applications of the results to constructing efficient computations in enumeration of various kinds of maps will be given.

(Muller): Cluster algebras were originally introduced to axiomatize part of the "dual canonical basis" for the ring of functions on a Lie group, and similar patterns have been found in many other contexts. This naturally begs the question: do cluster algebras have a natural basis, extending the set of cluster monomials?

Recent work by Gross, Hacking, Keel, and Kontsevich has proposed a basis of "theta functions". These theta functions are defined by counting certain tropical curves in a "scattering diagram", a task which appears to be prohibitively difficult in practice. In this talk, I will demonstrate how to use this counting problem to restrict the behavior of the theta functions; in particular, to constrain their monomial support. In simple cases (that is, rank 2), such constraints completely characterize the theta functions, and may be used to give alternate characterizations which are far more computable in practice. Parts of this talk will cover joint work with Man Wai Cheung, Mark Gross, Gregg Musiker, Dylan Rupel, Salvatore Stella, and Harold Williams.

(Pylyavskyy): Zamolodchikov periodicity is a property of certain discrete dynamical systems associated with quivers, called T-systems. One can consider an affine analog of those systems, which instead of periodicity exhibit certain forms of integrability. In particular, when one considers tropical version of those T-systems, affine slices of the quiver behave like solitons. We prove this in type A and provide a complete classification of T-systems where we expect this phenomenon. This is joint work with Pavel Galashin.

(Williams): We study the Newton-Okounkov bodies for the Grassmannian associated to plabic graphs G, and give a combinatorial formula for their lattice points in terms of the combinatorics of Young diagram. We also give an interpretation of them in terms of the superpotential for the mirror Grassmannian.

(Shapiro): In the first talk we give an elementary overview of hyperbolic approach to cluster algebra theory. Starting from the special coordinates on the configuration space of n points on $RP^1$ we define similar coordinates for the Teichmuller space $T_{g,s}$ of (decorated) Riemann surfaces of genus $g$ with $s$ punctures, discuss (compatible) Weil-Petersson Poisson bracket (pre-symplectic form) and then generalize the construction to two dual versions of (generalized) cluster algebras of Fomin and Zelevinsky together with the compatible Poisson bracket (pre-symplectic form). The talk is based on papers by Fock-Goncharov, Fomin-Zelevinsky, Gekhtman-S.-Vainshtein, Fomin-S.-Thurston, and S.-Chekhov.

(Chekhov): We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$ bordered cusps located on boundaries of holes uniformized by Poincare with the character variety of $SL(2,R)$-monodromy problem. The effective combinatorial description uses the fat graph structures; we can construct all observables, which are geodesic functions of closed curves and $\lambda$-lengths of paths starting and terminating at bordered cusps decorated by horocycles, out of extended shear coordinates; the Poisson and quantum algebras of observables are then induced by the Poisson and quantum structures of the extended shear coordinates; for $\lambda$-lengths we obtain the quantum cluster algebras of Berenstein and Zelevinsky. A seed of the corresponding quantum cluster algebra corresponds to the partition of $\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides are cluster variables constituting a seed of the algebra; their number $6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the dimension of $SL(2,R)$-character variety given by $[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$, where $B_i$ are Borel subgroups associated with bordered cusps. Moreover, we construct quantum $SL(2,R)$- and $SL(2,C)$-monodromy matrices themselves; the corresponding Poisson and quantum algebras can be presented in an R-matrix form; they coincide with the Fock-Rosly algebras of monodromies. The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov