# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45

**Organizers:** Franz Pedit, Ulrich Pinkall, Iskander A. Taimanov, Alexander Veselov, Katrin Wendland

## Monday, January 16

15:00 Leonid Rybnikov: Laumon spaces and Yangians

## Wednesday, January 18

10:00 Giovanni Felder: Gaudin subalgebras and stable rational curves

## Tuesday, January 31

10:00 Alexander Alexandrov: From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators

Abstract: In this talk we present our conjecture on the connection between the Kontsevich-Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the GL(∞) group element.

The important feature of this group element is its simplicity: it is built of only generators of the Virasoro algebra. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich-Witten tau-function.

11:30 Leonid Chekhov: Beta-ensembles and quantum Riemann surfaces

Abstract: Beta-ensembles, being generalizations of matrix models, enjoy a generalized "topological expansion" in the double-expansion form in two parameters: 1/N^{2} and ħ=(√β-√β^{-1})/N. (These two parameters are related to the Nekrasov-Shatashvili deformation parameters in such a way that the first is the product ε_{1}ε_{2} and the second is the difference ε_{1}-ε_{2}.) It is known since 2006 how to produce this double-series expansion for any potential whose derivative is a rational function. A more interesting and intriguing problem is to develop just a single-series expansion in 1/N^{2} keeping the second parameter, ħ, finite. It gives rise to what was called "quantum Riemann surfaces" in papers by L.Ch. B.Eynard and O.Marchal in which the polynomial case was considered. I will present new results obtained in collaboration with B.Eynard and S.Ribault on quantum Riemann surfaces for logarithmic potentials which, due to the AGT hypothesis, coincide with conformal blocks of (quantum) Liouville theory.

## Thursday, February 2

15:00 Piotr Grinevich: Virasoro action on finite-gap solutions of KdV and KP hierarchies

## Monday, February 6

10:00-12:00 Leonid Chekhov: Gaussian ensembles, discretized moduli spaces and Harer-Zagier recursion relations

## Wednesday, February 29

15:00-16:00 Nobutaka Boumuki: The classification of real forms of simple irreducible pseudo-Hermitian symmetric spaces

Abstract: The main purpose of this talk is to classify the real forms of simple irreducible pseudo-Hermitian symmetric spaces. That provides an extension of Jaffee's results (Bull. Amer. Math. Soc. 1975; J. Differential Geom. 1978), Leung's result (J. Differential Geom. 1979) and Takeuchi's result (Tohoku Math. J. 1984) concerning the classification of real forms of irreducible Hermitian symmetric spaces.

## Tuesday, March 6

14:30-16:00 Sebastian Heller: A spectral curve approach to Lawson's genus 2 surface

Abstract: In this talk we will discuss the family of flat connections associated to Lawson's minimal surface of genus 2. There exists a 1:2 correspondence between these flat connections and flat connections on a line bundle defined over a 2-torus. This enables us to parametrize the family of flat connections in terms of a spectral curve.

## Wednesday, March 7

14:30-16:00 Jorge Lira: Isometric immersions into Lie groups

Abstract: We plan to discuss some results about integrability conditions for the existence of isometric immersions into nilpotent and solvable Lie groups. As an application we present a generalized Weierstrass representation for surfaces immersed in Heisenberg space with prescribed mean curvature. If time permits we will comment about the same kind of result concerning minimal surfaces into Berger spheres.

## Monday, March 12

14:30-16:00 Andrei Marshakov: Integrable systems, cluster variables and dimer models

Abstract: I describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups, numerated by cyclic-irreducible elements of the co-extended affine Weyl groups. The phase spaces of these systems admit cluster coordinates and the simplest example of this construction for symplectic leafs in simple Lie groups gives rise to the relativistic Toda chain. This class of integrable systems turns out to coincide with those constructed out of dimer models on a two-dimensional torus and classified by Newton polygons in the work of Goncharov and Kenyon. I am going to discuss a few explicit examples. (The talk is based on joint work with Vladimir Fock.)

## Wednesday, March 14

14:30-16:00 Anton Zabrodin: Classical tau-function for quantum spin chains

Abstract: For any generalized quantum integrable spin chain we introduce a "master T-operator'' which is a sort of generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bi-linear equations of the Hirota form for the master T-operator, which allows one to identify it with tau-function of an integrable hierarchy of classical soliton equations. In this talk we will mostly concentrate on spin chains with rational GL(N)-invariant R-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic) R-matrices and to supersymmetric spin chains.

## Wednesday, March 21

14:30-16:00 Rei Inoue Yamazaki: Tropical geometry and integrable systems

Abstract: I will give an introduction to the theory of tropical curves and apply this to solve an integrable cellular automaton called the box-ball system. (Reference: arXiv:1111.5771.)

## Friday, March 23

11:00-12:30 Anton Izosimov: Stability in bihamiltonian systems

Abstract: I will tell how to use a bihamiltonian structure to simplify stability analysis for an integrable system. Stability of stationary rotations of a multidimensional rigid body and stability of finite genus solutions of periodic KdV will be two particular examples.

15:00-16:00 Hironao Kato: Flat projective structures, prehomogeneous vector spaces and castling transformations

Abstract: I will talk about the existence problems of invariant flat projective structures on homogeneous spaces, including a history and basic definitions. Especially, I will explain an important correspondence: from a given prehomogeneous vector space which is a certain representation considered by Sato Mikio, we can construct a projectively flat connection.

## Wednesday, April 11

14:30-16:00 Claus Hertling (Mannheim): Painlevé III, its isomonodromic connection as integrable twistor structure, and the geometry of the moving poles