Kohnert polynomials are a common generalization of Schubert polynomials and Demazure characters for the general linear group defined recently by myself and Searles. Demazure crystals are certain truncations of normal crystals whose characters are Demazure characters. For each diagram satisfying a southwest condition, we construct a type A Demazure crystal whose character is the Kohnert polynomial for the given diagram, resolving a conjecture that these polynomials expand nonnegatively into Demazure characters. I'll give explicit formulas for the expansions with applications including a bijective proof of Kohnert's rule for Schubert polynomials.

I will review some recent progress on quantum symmetric pair coideal subalgebras of quantum groups, in particular the construction of the universal K matrix, which bears significant resemblance to the construction of the universal R-matrix for the quantum group.

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation equals to the series of c=1 Virasoro conformal blocks. We study similar series of c=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blow-up relations on Nekrasov partition functions. We also study series of q-deformed c=−2 conformal blocks and relate it to q-Painlevé equation. Using this we prove formula for the tau function of q-Painlevé A(1)′_7 equation Based on joint work with A. Shchechkin.

Classically, we express symmetric polynomials In terms of Schur ones, but this cannot be the best, especially if you need to expand theta functions and the Kac-Moody characters. And of course we need some canonical bases in all polynomials (not only symmetric)! Presumably level-1 Demazure characters and relatively new level-1 thick (“upper”). Demazure characters are just fine. They are the key in nil-Daha theory (the limits t=0 and at infinity), and provide the characters of local Weyl modules and so-called nonsymmetric global Weyl modules (E.Feigin, Kato, Macedonskiy, … developing). Furthermore, they generalize classical q-Hermite polynomials and serve perfectly theory of Rogers-Ramanujan sums. Concerning the latter, we will connect it with a new version of 2d TQFT, extending the usual one to the Kac-Moody setting (if this is finished before the conference).

Work of de Gier and Nichols explored the two-boundary Temperley-Lieb algebra as a natural generalization of the classical Temperley-Lieb algebra from a statistical mechanics perspective. They present the two-boundary Temperley-Lieb algebra both as a diagram algebra and as a quotient of the affine Hecke algebra of type C. In work with I. Halacheva, A. Ram, and A. Wilbert, we have expanded upon connections both to the two-poled braid group (via the type-C affine Hecke algebra) and to generalized exotic Springer fibers. In this talk, we will explore some of the beautiful representation theoretic and combinatorial structure of these algebras.

Quantum cluster algebras are quantizations of cluster algebras, which are a class of algebras interpolating between integrable systems and combinatorics. These algebras were originally introduced to study positivity phenomena arising in the study of quantum groups, and so one of the key questions regarding them (and their quantum analogues) is whether they admit a basis for which the structure constants are positive. The classical version of this question was settled in the affirmative by Gross, Hacking, Keel and Kontsevich. I will present a proof of the quantum version of this positivity, due to joint work with Travis Mandel, based on results in categorified Donaldson-Thomas theory obtained in joint work with Sven Meinhardt.

Alternating Sign Matrices (ASM) are at the confluent of many interesting combinatorial/algebraic problems: Laurent phenomenon for the octahedron equation, configurations of the Square Ice (Six Vertex model), Descending Plane Partitions (DPP), etc. Here we consider the Triangular Lattice version of the Ice model with suitable boundary conditions leading to an integrable 20 Vertex model. Configurations give rise to generalizations of ASM, which we coin Alternating Phase Matrices (APM). We generalize the ASM-DPP correspondence by showing that APM are equinumerous to the quarter-turn symmetric domino tilings of a quasi-Aztec square with a central cross-shaped hole, and obtain a compact determinant formula for their enumeration. We also present conjectures for triangular Ice with other types of boundary conditions, and preliminary results on the shape of large APM. (joint work with E. Guitter, IPhT Saclay, France).

(Joint work with Pierre Goussard). We introduce a formalism to deal with formulas in classical Riemann geometry involving Levi-Civita connection, Einstein equation, Bianchi identity etc. The basic ingredients are forms with values in a Clifford algebra and the action of the sl(2) x sl(2) Lie algebra thereon. For the Kähler case this algebra generalises to the affine sl(4).

The talk will be based on the joint work with M. Bershtein and A. Marshakov. I will explain the general construction, which starts from arbitrary Newton polygon Δ and gives non-autonomous version of the discrete flow in corresponding cluster integrable system. General solution of this flow is given conjecturally in terms of 5D Nekrasov partition functions with q=t,corresponding to the same polygon Δ. We consider two families of Newton polygons: one family gives q-Painleve equations, another one – non-autonomus Hirota bilinear equations. We also construct quantiztion of such discrete flow and conjecture that in general situation its solution is given in terms of Nekrasov partition functions with arbitrary deformation parameters t and q.

In 2006 Zinn-Justin observed that the "puzzle" rule for Grassmannian equivariant Schubert calculus [K-Tao '03] is based on an R-matrix. We extended this to discover and prove puzzle rules for the multiplication on 2- and 3-step flag manifolds [K-ZJ '17], using R-matrices for representations of D_4 and E_6 respectively. We trace the utility of R-matrices for d-step flag manifolds to Maulik-Okounokv's construction of R-matrices in the equivariant homology of quiver varieties, thereby inspiring further rules. This work is joint with Paul Zinn-Justin.

We construct sub-coalgebras in the ring of symmetric functions (viewed as a Hopf algebra) that are distinguished by positivity. That is, these sub-coalgebras are spanned by a particular basis, called cylindric symmetric functions, which have non-negative integral expansion coefficients with respect to the coproduct. These cylindric symmetric functions arise in the context of partition functions of integrable lattice models and the resulting expansion coefficients have geometric and representation theoretic interpretations. In the case of cylindric Schur functions one obtains the Gromov-Witten invariants of Grassmannians and in the case of cylindric complete symmetric functions sums over tensor multiplicities of the generalised symmetric group.

The XXZ spin-1/2 chain at finite temperature T can be studied by means of the quan- tum inverse scattering method. In this approach, the physically pertinent observables at finite T, such as the per-site free energy or the correlation lengths, have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non-linear integral equations. The derivation of such representations was based on numerous conjectures: the possibility to exchange the infinite volume and the infinite Trotter number limits, the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the existence and uniqueness of solutions to the auxiliary non-linear integral equations, as well as the possibility to take the infinite Trotter number limit on their level. I will present a setting allowing one to prove all these conjectures for temperatures large enough. This is a joint work with F. Göhmann, S. Goomanee and J. Suzuki.

Stanley’s symmetrized chromatic polynomial, which generalizes the conventional chromatic polynomial, was discovered in middle 90’ies independently by R. Stanley on one side and S. Chmutov, S. Duzhin, and S. Lando (under the name of weighted chromatic polynomial) on the other side. We show that the generating function for the symmetrized chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev–Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants giving the same solution of the KP. The key point here is a Hopf algebra structure on the space spanned by graphs and the behavior of the invariants on its primitive space. It is interesting that similar Hopf algebras of other combinatorial objects seem to be not related to integrable hierarchies of partial differential equations. The talk is based on a joint work with S. Chmutov and M. Kazarian arXiv:1803.09800.

TBA

A number of combinatorial models describe the tensor product multiplicities for a complex semisimple Lie algebra in terms of the number of integer points in a convex polytope. Notable examples include the i-trail polytopes of Berenstein and Zelevinsky and the hive polytopes of Knutson and Tao. In the semiclassical limit, the multiplicities are approximated by the volumes of these polytopes, or equivalently by the symplectic volumes of corresponding reduced phase spaces. This talk will discuss various methods for computing tensor product multiplicities from such volume data, which in a sense amounts to recovering a quantum object from a classical one. We will focus on the paradigmatic case of Littlewood-Richardson coefficients, highlighting connections to symplectic geometry and random matrix theory.

In this talk I will consider the ODE/IM correspondence for all states of the quantum g-KdV model, where g is a simply laced affine Kac-Moody algebra. I will show how to construct quantum g-KdV opers as an explicit realization of a class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at 0 and infinity, and by allowing an arbitrary but finite number of additional singular terms with trivial monodromy. The generalized monodromy data of the quantum g-KdV opers satisfy the Bethe Ansatz equations of the quantum g-KdV model. In the sl2 case, the opers obtained are equivalent to the Schroedinger operators with "monster potential" obtained by Bazhanov, Lukyanov and Zamolodchikov in relation with the higher states of the quantum KdV model. Talk based on joint work with Andrea Raimondo.

It will be shown that differential equations for limit shapes of the 6-vertex model possess infinitely many conserved quantities which Poisson commute with respect to natural Poisson brackets. This property also holds for an inhomogeneous case.

A class of infinite dimensional quantum group representations, termed Gelfand-Zeitlin modules, were introduced in the 2000’s by Gerasimov, Kharchev, Lebedev and Oblezin. These representations and their generalizations have attracted recent interest due to their role in the theory of the quantized K-theoretic Coulomb branch algebras constructed by Braverman, Finkelberg and Nakajima. In my talk I will explain how the Whittaker spectral transform for the q-difference open Toda chain can be used to embed a large class of BFN Coulomb branch algebras into quantum cluster algebras. In particular, we obtain a novel description of the cluster structure of the quantum group U_q(sl_n) and its Gelfand-Zeitlin integrable system, which leads to a proof of the multiplicity-one branching rule for the principal series of U_q(sl_n,R).

Quantized higher Teichmüller theory, as described by Fock and Goncharov, assigns an algebra and its representation to a surface and a Lie group. This assignment is conjectured to give an (infinite dimensional) analog of a modular functor: in particular it should be local with respect to the operation of cutting surfaces. Moreover, the cutting is expected to be invariant under the mapping class group action. In this talk I will briefly remind the Fock–Goncharov construction, and outline the proof of locality. I will then focus on the mapping class group invariance, which involves computation of quantum monodromies. The latter is closely related to several known quantum integrable systems, as well as interesting algebraic combinatorics. This talk is based on joint works with Gus Schrader.

5 years ago, at the Yuri Stroganov memorial conference in this very same place, we presented the first application of the Tangent Method. The scope of this method is to determine the Arctic Curves of certain Integrable Systems, and our application was the 6-Vertex Model (6VM) with DWBC, at the ``combinatorial point'' Δ=-1/2. In this talk we try to present the various evolutions of this method. In particular, we discuss what we have found so far in trying to apply the method to the full phase diagram of the 6VM. Remarkably, for a large portion of the phase diagram, controlling the leading asymptotics of the 1-point boundary correlation function is sufficient for determining the Arctic Curve rigorously. Work in collaboration with F.Colomo (U. Florence)

The isomonodromic tau-functions are usually only defined up to monodromy-dependent normalisation factors. Exact WKB can be used to define preferred normalisations completing the definition of the tau-functions. An important role is played by certain coordinates for the character variety which represent solutions to Riemann-Hilbert problems similar to those considered by Gaiotto-Moore-Neitzke and Bridgeland in the context of BPS-invariants.

I will show that the Hilbert series of the projective variety X=ℙ(O_min), corresponding to the minimal nilpotent orbit O_min, is universal in the sense of Vogel. Namely, I will explain that it can be written uniformly for all simple Lie algebras in terms of Vogel's parameters α,β,γ and represents a special case of the generalized hypergeometric function ₄F₃. The proof follows from Borel-Hirzebruch description of the coordinate ring of X as 𝔤-module and the results of Landsberg and Manivel. A universal formula for the dimension and degree of X is then deduced.

Baxter developed a method for finding the spectrum of the Hamiltonian of the closed XYZ Heisenberg chain. It involves the construction of a family of operators Q(y) in addition to the usual transfer matrices T(z). All these operators pairwise commute and crucially there is also a functional equation (``Baxter's TQ relation'') from which Bethe ansatz equations can be derived. In work in the 90s by Bazhanov, Lukyanov, Zamolodchikov this formalism was placed in the context of representations of affine quantum groups and in particular their standard Borel subalgebras. We discuss such a formalism for the open XXZ chain with diagonal boundaries; it involves certain representations of quantum affine sl2 and certain subalgebras. The operators Q(y) and T(z) are given by a generalized version of Sklyanin's two-row construction. The key tools for proving the functional equation are short exact sequences of modules of the two standard Borel algebras and a well-chosen algebra automorphism of quantum affine sl2 interchanging them. Work in progress, joint with Robert Weston.

In this talk we explain an interpretation of the Kasteleyn operator of a doubly-periodic bipartite graph from the perspective of homological mirror symmetry. Specifically, given a consistent bipartite graph G in T^2 with a complex-valued edge weighting E we show the following two constructions are the same. The first is to form the Kasteleyn operator of (G,E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in (C*)^2. The second is to take a certain Lagrangian surface L in T^* T^2 canonically associated to G, equip it with a brane structure prescribed by E, and pass to its homologically mirror coherent sheaf. This lives on a toric compactification of (C*)^2 determined by the Legendrian link which lifts the zig-zag paths of G (and to which the noncompact Lagrangian L is asymptotic). As a corollary, we obtain a complementary geometric perspective on certain features of algebraic integrable systems associated to lattice polygons, studied for example by Goncharov-Kenyon and Fock-Marshakov. This is joint work with David Treumann and Eric Zaslow.