Program

	4 (Mon)	5 (Tue)	6 (Wed)	7 (Thu)	8 (Fri)	9 (Sat)
10:00~11:00	Bazhanov	Chari	Feigin	Matveev	Molev	van Diejen
11:10~12:10	Katsura	Loktev	Mukhin	Takhtajan	Noumi	Hikami
(lunch)
13:40~14:40	Foda	Toledano Laredo	Takeyama	Reshetikhin	Buch	Warnaar
14:50~15:50	Ip	Konno	Kuwabara	Naoi	Lando	Ishikawa
(break)
16:20~17:20	Nagoya		Miwa*	Kirillov*	Maeno	Nakanishi

* Special lectures are from 16:00 ~ 17:30.

Titles and Abstracts

Vladimir Bazhanov: Yang-Baxter equation: ten steps forward

The Yang-Baxter equation is a key mathematical concept in the theory of integrable systems in statistical mechanics and quantum field theory. In this talk I will review the most important steps in our understanding of the Yang-Baxter equation as well as its connections to other fields over the past thirty years. This includes connections to quantum groups and algebras, dynamical evolution systems, integrable systems in three dimensions, theory of elliptic hypergeometric functions and quantum geometry. In conclusion I will discuss some outstanding and challenging mathematical problems in this field.

Anders Buch: Curve neighborhoods

Given a generalized flag manifold X = G/P, a Schubert variety X(w), and a degree d, consider the set of points that can be reached from X(w) by a rational curve of degree d, i.e. the union of all rational degree d curves through X(w). It turns out that the Zariski closure of this set is a larger Schubert variety, which is important for many aspects of the quantum cohomology of X, including the quantum Chevalley formula and the smallest q-degree in the quantum product of two Schubert classes. I will give a very explicit description of this "curve neighborhood" of the Schubert variety in terms of the Hecke product of Weyl group elements, and use it to give a simple proof of the (equivariant) quantum Chevalley formula. This is joint work with Leonardo Mihalcea.

Vyjayanthi Chari: On the category of graded integrable modules for the current algbera associated to a simple Lie algebra

We begin by explaining briefly the connection between the representation theory of quantum affine algebras and the current algebras. We shall see that it is natural to restrict our attention to the graded integrable modules for the current algebra. This category is not semisimple and has many similarities with other well--known non--semisimple categories in Lie theory. The global Weyl modules play the role of standard modules in this category. The co-standard modules are the dual local Weyl modules and we shall focus on understanding the full subcategory of tilting modules: objects which have both a standard and a co-standard filtration.

Boris Feigin: Extension of vertex operator algebras of AGT type

Omar Foda: Scalar products in integrable spin chains and 3-point functions in N=4 super Yang-Mills

I wish to review some recent results on Slavnov-type scalar products in su(2) and su(3) Heisenberg spin-chains that were motivated by applications in maximally supersymmetric　4-dimensional Yang-Mills theory.

Kazuhiro Hikami: On the complex volume of knots

It is known that the complex volume of knots is related to asymptotics of the colored Jones polynomial. I discuss some aspects of the complex volume. This talk is based on a joint work with R.Inoue.

Ivan C.H. Ip: Positive Representations of Split Real Quantum Groups

In this talk, I will introduce the family of positive principal series representations for split real quantum groups by positive self-adjoint operators. The construction of these representations gives the starting point of a new research program devoted to the representation theory of split real quantum groups initiated in the joint work with Igor Frenkel. It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finite-dimensional representation theory of quantum groups.　Recently from the construction of the positive representations, a direct analytic relation between modular duality and Langlands duality is also discovered, which should have deep consequences in the Langlands program. The universal R operator in the context of positive representation is also recently obtained via the language of multiplier Hopf algebra.

Masao Ishikawa: (q,t)-hook formula for Birds and Banners

We study Okada's conjecture on multivariate hook formula for d-complete posets. We give a proof of the (q,t)-hook formula for Birds and Banners using the Macdonald polynomials and Gasper's identity for very well poised series ${}_{12}W_{11}$.

Hosho Katsura: Inhomogeneous but solvable/integrable models

I will present two classes of interacting one-dimensional systems that are seemingly unrelated to the models solvable by the standard techniques such as the Bethe ansatz and the Yang-Baxter relation. An example of the first class is the XY spin chain with sine-square deformation. The model is defined on an open chain and the local Hamiltonians are modified according to the sine-square function. Due to this inhomogeneity, the single-particle eigenstates cannot be obtained in closed form. However, I will show that the many-body ground state can be obtained exactly and it is identical to the ground state of the uniform and periodic XY chain which can be solved by the free-fermion method. The same correspondence holds for the critical quantum Ising chain and more general conformal field theories. In the second class of examples, a system is made up of spins and fermions on a zigzag ladder. The Hamiltonian of the system is defined as the anticommutator of two nilpotent ``supercharges". These charges together with the Hamiltonian form the supersymmetry algebra. I will show that the spectrum of the Hamiltonian exhibits a number of fascinating properties. For example, the degeneracy of each energy level of the chain of length $2L$ is $2^L$ even in the presence of spatially varying couplings. I will argue that the model possesses an infinite dimensional symmetry algebra that is quite reminiscent of the Yangian in the Haldane-Shastry model.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] H. Katsura et al., in preparation.

Hitoshi Konno: Elliptic quantum groups, quantum Z-algebras and deformed W-algebras

After making some general remarks on the face type, i.e. dynamical, elliptic quantum groups, we define two elliptic algebras $U_{q,p}(\hat{g})$ and $E_{q,p}(\hat{g})$ as a certain topological algebras in the $p$-adic topology. $U_{q,p}(\hat{g})$ and $E_{q,p}(\hat{g})$ can be regarded as elliptic analogues of the quantum affine algebra $U_{q}(\hat{g})$ in the Drinfeld realization and in the FRST formulation, respectively. In the case $\hat{gl}_N$, we show that they are isomorphic. Then we discuss the infinite dimensional highest weight dynamical representations of $U_{q,p}(\hat{g})$. We define a quantum dynamical analogue of Lepowsky-Wilson's $Z$-algebras and give a general structure of the irreducible $U_{q,p}(\hat{g})$-modules obtained through the $Z$-algebras. After giving some examples on the level-1 irreducible representations we discuss some conjectures that there exists a deformation of the $W$-algebras associated with the coset $\hat{g}\oplus \hat{g}\supset (\hat{g})_{{\rm diag}}$ with level $(r-g-k,k)$ ($g$:the dual Coxeter number),which contains Fateev-Lukyanov's $WB_l$-algebra, and that it acts on the level-$k$ $U_{q,p}(\hat{g})$-module.

Toshiro Kuwabara: BRST cohomologies for rational Cherednik algebras

Sergei K. Lando: On computation of universal polynomials for characteristic classes of singularities

Any space of meromorphic functions is stratified according to the number of critical values of these functions. The cohomology classes Poincar\'e dual to the strata of such a stratification can be described in terms of certain universal classes. The standard tool for the description are Thom polynomials. Their computation is a complicated problem, which usually is done step-by-step and requires the knowledge of a complete classification of singularities.

However, in some cases computation of characteristic classes for spaces of meromorphic functions on algebraic curves can be done explicitly for large series of classes. Certain corresponding generating functions are solutions to integrable hierarchies. Further computations pose interesting problems of both geometric and combinatorial nature.

The talk is based on a joint work with Maxim Kazarian.

[KL1] M. Kazaryan, S. Lando, Towards an intersection theory on Hurwitz spaces, RAS Izv., Mathematics, v.68, 935--964 (2004).

[KL2] M. Kazaryan, S. Lando, Topological Relations on Witten--Kontsevich and Hodge Potentials, Moscow Mathematical Journal, v. 12. no.~2. pp. 397--411 (2012).

[KLZ] M. Kazaryan, S. Lando, D. Zvonkine, New topological recursion for the genus zero Hurwitz numbers, in preparation.

Sergey Loktev: Character of Weyl modules and proof of BGG duality for current algebras

Weyl modules are defined as highest weight representations of current algebras (including multi-variable case) satisfying universality property. We summarize, what is known for their characters, and complete the proof of the BGG duality (presented in V. Chari's talk) for the one-variable polynomial type A currents.

Toshiaki Maeno: The Fomin-Kirillov quadratic algebra and its affinization

In the early '90s, Fomin and Kirillov introduced a noncommutative quadratic algebra to give a combinatorial description of the cohomology ring of the flag variety of type A. Their construction shows an interesting connection between the Schubert calculus and integrable systems. It is remarkable that their algebra admits a natural quantum deformation that is compatible with the structure of the quantum cohomology ring of the flag variety. In this talk, I will survey some known results on the Fomin-Kirillov quadratic algebra and introduce its affine version. In particular, the quantum deformation of the algebra can be understood from the braided differential calculus on the affine Weyl group.

Vladimir Matveev: Large parametric asymptotic of the multi-rogue waves solutions of the NLS equation and extreme rogue wave solutions of the KP-I equation

In this talk, we present some new still unpublished results concerning the behavior of the multi-rogue waves solutions of the focusing NLS and KP-I equation. These results are based on explicit polynomial formulas obtained from the determinant representations for these solutions found in my works with Philippe Dubard [1]--[2]. In these works the concept of the multiple rogue waves solutions both for focusing NLS equation and KP-I equation was first introduced. These works provided an explanation of the fact that the so called higher Peregrine breathers (we'll call them for brevity $P_{n}$ breathers) with $n \geq 2$ are not isolated and correspond to particular choice of parameters for the rank $n$ multi-rogue wave solution depending on 2n free real parameters. In 2010 only genuine Peregrine breather (i.e. $P_{1}$ breather), $P_{2}$-breather (found in 1995 by Akhmediev, Eleonski and Kulagin) and $P_3$ breather (found in 2009 by Akhmediev, Ankiewicz and Soto-Crespo) were known explicitly. The discovery of the multiple rogue-waves solutions stimulated the study of their particular cases corresponding to the different choices of parameters revealing quite different symmetric and asymmetric configurations. It seems that a rigorous study of the large parametric behavior of the multi-rogue wave solutions was never performed before. In particular, we will show that (at least for small ranks) that all multiple rogue wave solutions of the rank $m\leq n-2$ can be obtained as an appropriately chosen large parametric limits of the rank $n$ solutions.

[1] P. Dubard, P. Gaillard, C. Klein and V. Matveev, On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation, Eur. Phys. J. Spec. Top., 185, 247--258 (2010).

[2] P. Dubard and V. Matveev, Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation, Nat. Hazards Earth Syst. Sci., 11, 667--672 (2011).

Alexander Molev: Feigin-Frenkel center and Yangian characters

For each simple Lie algebra g consider the vacuum module V(g) at the critical level over the corresponding affine Kac-Moody algebra. The vacuum module has a vertex algebra structure. We construct explicit generators of the center of this vertex algebra for each Lie algebra g of classical type. This leads to a new proof of the Feigin-Frenkel theorem (1992) and to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g). Moreover, we use Yangian characters (or q-characters) of Kirillov-Reshetikhin modules to calculate the images of the central elements under an affine version of the Harish-Chandra isomorphism.

Evgeny Mukhin: On representations of quantum toroidal gl(n)

We will discuss tame representations of quantum toroidal gl(n). We will show that many such representations can be constructed explicitly, and that the study of their structure and characters leads to interesting combinatorics involving plane partitions with various boundary conditions.
As an application we obtain a combinatorial description of various modules of various algebras including affine gl(n), gl(\infty), W_n. This is a report on a joint ongoing project with B. Feigin, M. Jimbo and T. Miwa.

Hajime Nagoya: From Gauss to quantum Painleve

A spectral type A is a tuple of partitions of a positive integer N. I propose a conjecture that a family of hypergeometric integrals associated with a specral type A gives a Schroedinger equation whose classical limit is a Hamiltonian system describing isomonodromy deformation for the Fuchsian system associated with the spectral type A. I discuss several examples starting with the quantum sixth Painleve equation.

Tomoki Nakanishi: Wonder of sine-Gordon Y-systems

The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and only a part of them have been proved so far. We formulate these Y-systems by the polygon realization of cluster algebras of types A and D, and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems.

This is a joint work with Salvatore Stella.

Katsuyuki Naoi: An approach to the X=M conjecture using current algebras

A one-dimensional sum is a weighted sum over the highest weight elements of a tensor product of Kirillov-Reshetikhin crystals, and the X=M conjecture asserts that a one-dimensional sum has some explicit formula called the fermionic formula. This conjecture has been proved in several cases using combinatorial methods, for example in type A by Kirillov, Schilling and Shimozono. In this talk I will introduce a new approach to this conjecture using the representation theory of the current algebra $\mathfrak{g}\otimes\mathbb{C}[t]$ associated with a simple Lie algebra $\mathfrak{g}$, which gives a proof of the conjecture in type A and D.

Masatoshi Noumi: An elliptic extension of Askey-Wilson polynomials and associated elliptic Schur functions

In this talk I introduce a family of elliptic functions that generalize Askey-Wilson polynomials, and discuss their fundamental properties including symmetries and difference equations. Also, I investigate a class of multivariable elliptic functions (of Schur type) built up from them by determinants. This class of functions can be regarded as an elliptic extension of Koornwinder polynomials with t=q, and carries various characteristic properties such as spectral duality and binomial formula.

Nicolai Yu. Reshetikhin: Bethe vectors and solutions to the reflections q-KZ equation

Reflection qKZ equation describes correlation functions and form-factors in integrable systems with integrable boundary conditions. It corresponds to B, C, D qKZ systems in Cherednik's classification. We solve these equations for quantum affine $sl_2$ and diagonal reflections matrices using "off-shell" eigenvectors of transfer-matrices proposed by Sklyanin. The talk is based on a joint work with J. Stokman and B. Vlaar.

Yoshihiro Takeyama: A generalization of duality for finite multiple harmonic sum

In 2005 D. M. Bradley proved a duality for multiple harmonic q-series, which plays a key role in the proof of quadratic relations for a q-analogue of multiple zeta values. In this talk I give another proof for the duality by means of generating functions, and discuss a generalization to elliptic case.

Leon Takhtajan: Symplectic structure on the moduli of differential equations and complexified Liouville equation

The main object of the talk is the moduli space of Fuchsian differential equations with equal exponents on the Riemann sphere, considered as complex symplectic manifold. I will explain how one can explicitly evaluate its symplectic form in terms of the monodromy data, and to show that complex Fenchel-Nielsen coordinates are its canonical (Darboux) coordinates. When restricted to the real (Fuchsian) slice, and using our old results with P. Zograf on Liouville action, this yields a direct proof of S. Wolpert result that Fenchel-Nielsen coordinates are canonical coordinates for the Weil-Petersson symplectic form. Finally, for the general case I will introduce complexified Liouville equation, and explain that its classical action is a generating function of the canonical transformation to the monodromy data. This is a joint work with P. Zograf.

Valerio Toledano Laredo: From Yangians to quantum loop algebras via abelian difference equations

The finite-dimensional representations of the Yangian Y_h(g) and quantum loop algebra U_q(Lg) of a complex, semisimple Lie algebra have long been known to share many similar features. Assuming that q is not a root of unity, I will explain how to construct an equivalence of categories between finite-dimensional representations of U_q(Lg) and an appropriate subcategory of finite-dimensional representations of Y_h(g). This equivalence is governed by the monodromy of an additive, abelian difference equation.

This is joint work with Sachin Gautam.

Jan F. van Diejen: Orthogonality of Macdonald Polynomials with Unitary Parameters

For any admissible pair of irreducible reduced crystallographic root systems, we present discrete orthogonality relations for a finite-dimensional system of Macdonald polynomials with parameters on the unit circle subject to a truncation relation.

Ole Warnaar: Hall-Littlewood functions and characters of affine Lie algebras

In 2000 Anatol Kirillov extensively studied the combinatorics of the modified Hall-Littlewood polynomials, and showed how these polynomials are related to the characters of the level-1 basic representation of $\mathrm{A}_{n-1}^{(1)}$. In this talk I will try to explain how, more generally, the modified Hall--Littlewood polynomials may be used to give combinatorial formulas for characters of affine Lie algebras of $\mathrm{BC}_n$ type at arbitrary level.