Guangdong Technion - Israel Institute of Technology (GTIIT) is dedicated to fostering world-level mathematics discussions. This time, the chosen topic is integrable systems and related combinatorics questions. To this goal, we invited the leading experts on this topic from BIMSA to share their views and exchange knowledge of discrete and continuous integrable systems with the GTIIT community. The questions that we will discuss include Discrete Painlevé Equations, asymptotical combinatorics of Schur measures, unimodality and log-concavity of generalized Gaussian polynomials, toric degenerations and Newton--Okounkov bodies, hybrid integrable systems and  sl_2 weight systems on graph complexes.

This mini-conference will be hosted in GTIT onsite, north campus, on Thursday and Friday, 18-19 April.

Schedule

April 18, Thursday

14:00 (E408) Anton Dzhamay

Geometry of Discrete Integrable Systems: QRT Maps and Discrete Painlevé Equations

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 these 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 birational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic.

This talk is based on joint work with Stefan Carstea (Bucharest) and Tomoyuki Takenawa (Tokyo).

15:00 (E408) Pavel Nikitin 

Asymptotics of skew Howe duality

We plan to survey some results on asymptotics of Schur-Weyl type dualities. We will start with the classical result on asymptotics of symmetry types of tensors due to S. Kerov and Ph. Biane, recall briefly the Schur measures and then discuss the recent results on asymptotics of skew Howe duality.

16:15 (E409) Anatol Kirillov

On unimodality and log-concavity of some polynomials, and their generalizations

A wide variety of polynomials which appear in algebraic geometry, algebraic combinatorics, representation theory of Lie algebras, probability theory, etc. are polynomials with negative integer coefficients. 

In my talk, I consider one variable polynomials. Being originated from geometry, combinatorics or algebra, these polynomials have many special properties which are important to investigate and prove. Among such properties are unimodality, log-concavity / ultra log-concavity / log-convexity, stability, P-positivity, Lorenzianness, and so on. In my talk, I want to explain a combinatorial proof of unimodality and log-concavity of generalized Gaussian polynomials and to state some conjectures about the P-positivity of the characteristic polynomial of any (convex) matroid on the set [1, n].

18:00 (E510) Fedor Pavutnitskiy 

Contemporary computers in contemporary mathematics (colloquium)

I would like to give an overview of two emerging applications of computers in pure mathematical research: formal theorem proving and machine learning for mathematics. These topics are by no means new, but in the last years the corresponding toolset available to mathematicians and scientists expanded greatly, which led to a mass of new, often groundbreaking results. I will give an overview of the recent research in these areas and sketch some possible projects and directions.

April 19, Friday

10:00 (E209) Yevhen Makedonskyi

Poset polytopes and pipe dreams: types C and B

In types C and B we construct new families of toric degenerations and Newton--Okounkov bodies of flag varieties and also of PBW-monomial bases in irreducible representations. The constructed objects are given by certain poset polytopes which in both types form families interpolating between the respective Gelfand--Tsetlin and FFLV polytopes.I will explain the construction of these polytopes using the version of pipe dreams construction. This construction os the same in types B and C.

11:00 (E209) Ivan Sechin

Ruijsenaars duality in geometric picture

Ruijsenaars duality is a nontrivial correspondence of two classical integrable systems on the same phase space, which relates the conservation laws or action variables of one system with the coordinates of the other, and vice versa. This duality allows us to describe the behavior of these integrable systems solving algebraic equations instead of differential equations of motion.

The powerful tool to find and prove such dualities is based on the group-theoretical approach. We start with a big phase space with a group-theoretical origin with a natural Hamiltonian action of a Lie group. This action allows us to make a Hamiltonian reduction procedure, which connects this large phase space with the smaller one. On this big phase space, there exist two different sets of invariant Poisson-commuting functions, which become coordinates and action variables of integrable systems after the reduction.

My main example will be based on the cotangent bundle to Lie group G with the action of its maximal compact subgroup and its unipotent subgroup. The dual classical integrable systems that appear in this case are open Toda chains and Goldfish models, corresponding to the Lie algebra of G.

14:00 (E209) Maxim Karev 

Conjectural extension of the sl_2 weight system to the Lando graph bialgebra

Kontsevich's construction associates a weight system, i.e a function on the algebra of chord diagrams modulo 4T relations, to any metrized Lie algebra. Kontsevich’s weight system takes values in the center of the universal enveloping algebra, which in the case of sl_2 is generated by a single element c. Chmutov and Lando have proved that sl_2 weight system takes the same value on any pair of chord diagrams that have the same intersection graph.

However, it is known that there are graphs that cannot be realized as intersection graphs of chord diagrams. So the natural question of the extension of the sl_2 weight system to the Lando graph bialgebra, that plays the role of graph-theoretic counterpart of the algebra of chord diagrams modulo 4T relations, arises. The Lando conjecture asserts that such an extension is unique, but for today not a single extension is known.

I will tell about the conjectural candidate for being the value of the extension of the sl_2 weight system to the Lando bialgebra at the point c = 3/8.

15:00 (E209) Liashyk Andrei 

Hybrid integrable systems 

We develop the framework for quantum integrable systems on an integrable classical background. We call them hybrid quantum integrable systems (hybrid integrable systems). They occur naturally in the semiclassical limit of quantum integrable systems.

I outline the concept of hybrid dynamical systems. Then I give several examples of hybrid integrable systems. The first series of examples is a class of hybrid integrable systems that appear in the semiclassical limit of quantum spin chains. Another example is the semiclassical limit of the quantum spin Calogero--Moser system. The result is a hybrid integrable system driven by usual classical Calogero--Moser (CM) dynamics. This system at the fixed point of the multi-time classical dynamics CM system gives commuting spin Hamiltonians of Haldane--Shastry model.

Text/Photos: GTIIT MCS Program, GTIIT News & Public Affairs