Abstracts
Marianna Russikikh , Playing dominos in different domains
We will discuss an extension of several results of Richard Kenyon on the dimer model. In 1999 he has shown that fluctuations of the height function of a random dimer tiling on Temperley discretizations of a planar domain converge in the scaling limit to the Gaussian Free Field with Dirichlet boundary conditions. We will discuss an extension of this result to other classes of discretizations. In particular, we will discuss boundary conditions of the coupling function in the so-called "even" domains. Interestingly enough, in this case the coupling function satisfies the same Riemann-type boundary conditions as fermionic observables in the Ising model. The main tool is a factorization of the gradient of the expectation of the height function in the double-dimer model into a product of two discrete holomorphic functions. In particular, we use this factorization to show that, rather surprisingly, the expectation of the double-dimer height function in the Temperley case is exactly discrete harmonic even before taking the scaling limit.
Anton Dzhamay, Geometry of Discrete Integrable Systems
Many interesting examples of discrete integrable systems can be studied from the geometric point of view. In this talk we will consider two classes of examples of such system: autonomous (QRT maps) and non-autonomous (discrete Painlevé equations). We introduce some geometric tools to study such systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and construction of a binational representation of affine Well symmetry groups that gives a complete algebraic description of our non-linear dynamic. If time permits, we also explain the relationship between this picture and classical differential Painlevé equations.
Nora Ganter, Examples of categorical groups
This talk gives an introduction to the theory of categorical groups, their representations and characters.
Alexander Shapiro, Positive representations of quantum groups
Positive representations are certain bimodules for a quantum group and its modular dual. In 2001, Ponsot and Teschner constructed these representations for U_q(\mathfrak{sl}_2) and proved that they form a continuous braided monoidal category, where the word "continuous" means that a tensor product of two representations decomposes into a direct integral rather than a direct sum. Ten years later, their construction was generalized to all other types by Frenkel and Ip. Although the corresponding categories were braided more or less by construction, it remained a conjecture that they are monoidal. Following a joint work in progress with Gus Schrader, I will discuss the proof of this conjecture for U_q(\mathfrak{sl}_n). The proof is based on our previous work where the quantum group is realized as a quantum cluster \mathcal X-variety. If time permits, I will outline a relation between this story and the modular functor conjecture in higher Teichmüller theory along with several other applications
Leon Takhtajan (Stony Brook), Symplectic geometry of the space of complex projective structures on orbifold Riemann surfaces
The space of complex projective connections on Riemann surfaces is an affine bundle, modeled on the holomorphic tangent bundle of the moduli space, and carries a holomorphic symplectic form - Liouville (2,0) form. The monodromy map is a holomorphic mapping of this space into the PSL(2,C)-character variety of an orbifold Riemann surface. The latter variety carries a natural symplectic form, introduced by Bill Goldman for compact for the compact surface. An old theorem (Kawai, 1996) states that in the compact case a pullback of Goldman symplectic form is the Liouville (2,0) form. However, Kawai’s prove was not very insightful. I will explain how one can determine the differential of the monodromy map and to compute the pullback directly, in the spirit of Riemann bilinear relations. This elucidates the symplectic structure of these varieties and proves a generalization of Kawai theorem - a pullback of Goldman symplectic form is the Liouville (2,0) form - for orbifold Riemann surfaces.
Semen Artamonov (Rutgers), Representation theory of genus two A_1 spherical DAHA
Spherical Double Affine Hecke Algebra can be viewed as a noncommutative (q,t)-deformation of the SL(N,C) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus SL(2,Z) acts by atomorphisms of DAHA. In my talk I will define a genus two analogue of A_1 spherical DAHA and show that the Mapping Class Group of a closed genus two surface acts by automorphisms of such algebra. I will then show that for special values of parameters q,t satisfying q^n t^2=1 for some nonnegative integer n this algebra admits finite dimensional representations. I will conclude with discussion of potential applications to TQFT and knot theory.
Xiaomeng Xu (MIT), Stokes phenomenon and its applications in mathematical physics
We will give a brief introduction to Stokes phenomenon of an ODE with irregular singularities and its relation with Poisson geometry and integrable systems. We then consider the Stokes phenomenon and isomonodromic deformation of Knizhnik–Zamolodchikov type equations, and explore their relations with Yang-Baxter equations and quantization of semisimple Frobenius manifolds.
Peter Koroteev, Quantum K-theory of quiver varieties and integrable systems
We shall define quantum K-theory of Nakajima quiver varieties using quasimap counts. For type A quivers we shall discuss in detail the connection with integrable systems of different types — integrable spin chains (XXZ) and many body systems of Ruijsenaars-Schneider type. These two types of models are related by so-called quantum/classical duality which we can explained geometrically.
Drazen Petrovic, Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice
We prove the Pfaffian Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifically, we prove that the Pfaffian of the Kasteleyn periodic-periodic matrix is negative, while the Pfaffians of the Kasteleyn periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. As an application, we obtain an asymptotics of the dimer model partition function with an exponentially small error term. This is a joint work with Pavel Bleher and Brad Elwood.
Kirill Vaninsky, Hamiltonian formalism for the finite open Toda lative and it's spectral curves
A Poisson formalism for integrable systems is a well-studied topic. Nevertheless,
there is no approach which will cover all known examples integrable with the machinery of
spectral curves. Very often there are no answers even to the simplest questions. We will formulate the main conjecture of the author regarding Poisson structures for these integrable systems. As an example, we consider the finite open Toda lattice.