at 17.00 via Zoom

**May 31, 2021
Oleg Alekseev **(Chebyshev Lab., S.-Petersburg State Univ.)

Stochastic Laplacian growth, II

Oleg Alekseev

Stochastic Laplacian growth, I

*2D Laplacian Growth (LG) is known as a universality class of diffusion driven growth processes, characterized by classical integrability connected to the dispersionless 2D Toda hierarchy. A discrete counterpart of LG is Diffusion-Limited Aggregation (DLA), where a point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. In the talk we discuss a stochastic growth model which provides a natural interpolation between LG and DLA. By using random matrices, we obtain a probability measure on the stochastic growth process. We show that the measure is closely connected to the 1-dimensional sigma model on the moduli space of smooth closed curves. Besides, by coupling conformal field theory (CFT) with stochastic LG we show that certain CFT correlation functions are martingales of the stochastic interface dynamics*

Igor Makhlin

The Gröbner fan of the flag variety

*The Gröbner fan of an ideal parametrizes its initial ideals and, therefore, the Gröbner degenerations of the corresponding algebraic variety. In particular, studying the Gröbner fan of the Plücker ideal is a natural approach to the problem of describing and classifying the flat degenerations of the flag variety. I will discuss the relatively little information we have about the structure of this fan and various open questions that arise here. The talk is based on joint work with Xin Fang, Evgeny Feigin and Ghislain Fourier*

Taras Panov

Complex geometry of manifolds with torus action

*Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds with a holomorphic torus action. A complex moment-angle manifold Z is constructed via a certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In this case, the invariants of the complex structure of Z, such Dolbeault cohomology and the Hodge numbers, can be analysed using the Borel spectral sequence of the holomorphic bundle.
In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the algebraic torus action. Examples of moment-angle manifolds include the Hopf manifolds, Calabi-Eckmann manifolds, and their deformations. The holomorphic foliated manifold (Z,F) has been also studied as a model for non-commutative toric varieties in the works several authors (arXiv:1308.2774, arXiv:1705.11110).
We construct transversely Kaehler metrics on moment-angle manifolds Z, under some restriction on the combinatorial data. We prove that all Kaehler submanifolds in such a moment-angle manifold lie in a compact complex torus contained in a fibre of the foliation F. For a generic moment-angle manifold Z in its combinatorial class, we prove that all its subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that Z does not have non-constant meromorphic functions, i.e. its algebraic dimension is zero.
Battaglia and Zaffran (arXiv:1108.1637) computed the basic Betti numbers for the canonical holomorphic foliation on a moment-angle manifold Z corresponding to a shellable fan. They conjectured that the basic cohomology ring in the case of any complete simplicial fan has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. We prove the conjecture. The proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on Z.
The talk is based on joint works with Hiroaki Ishida, Roman Krutowski, Yuri Ustinovsky and Misha Verbitsky*

Andrei Grekov

Infinite number of particles limit for the coordinate trigonometric degeneration of the Dell integrable system

*Using the non-commuting analog of the Cherednik operators for the coordinate trigonometric degeneration of the Shakirov-Koroteev Hamiltonians their N → ∞ limit will be constructed. Namely, we will apply the Nazarov Sklyanin idea to express the Hamiltonians in terms of the so-called covariant Cherednik operators, which preserve the space of polynomials symmetric in all variables except for one, thus having a well defined N → ∞ limit as operators on the space of polynomials in the formal variable with coefficients in symmetric functions. In the spin chains terminology, this space has a meaning of auxiliary space for the system, and the Hamiltonians will be expressed as matrix elements of the infinite number of some sort of L-operators
(Based on the paper https://arxiv.org/abs/2102.06853)*

Shamil Shakirov

Quantum DELL system and around

*It has been long expected that trigonometric Ruijsenaars-Schneider integrable system should admit a generalization elliptic both in coordinates and in momenta, the so-called double elliptic system or DELL for short. In this talk we review the recent development of 1906.10354 by P.Koroteev and the speaker, where a candidate for such a system has been put forward. An advantage of this proposal is that it is fully explicit at quantum level, i.e. formulated as a set of commuting difference operators. We cover the basic properties of the resulting system and its eigenfunctions, comment on relation to the previous proposals for DELL, and discuss relation to other double-elliptic constructions in mathematical physics*

Vadim Prokofev

Integrable hierarchies of nonlinear differential equations and many body systems. Elliptic case

Alexander Tsymbaliuk

Shifted Yangians and quantum affine algebras

*In the first part of the talk, I will recall some basic results about the shifted Yangians (and their trigonometric versions-the shifted quantum affine algebras). In type A and the when the shift is dominant, the shifted Yangians were originally introduced by Brundan-Kleshchev about 15 years ago who related their truncations to the type A finite W-algebras. A general framework was developed more recently, mostly motivated by the BFN (Braverman-Finkelberg-Nakajima) study of Coulomb branches. The exact relation is based on the generalization of GKLO (Gerasimov-Kharchev-Lebedev-Oblezin) homomorphisms.
In the second part of the talk, I will try to convince that the case of antidominant shifts (opposite to what was originally studied) is of particular importance as the corresponding algebras admit the RTT realization (at least in the classical types). As a consequence, one can translate the above GKLO-type homomorphisms into a wide class of type ABCD rational/trigonometric Lax matrices (some of which were known before, though they were scattered around in the various physics literature).
This talk is based on the joint works with M. Finkelberg, R. Frassek and V. Pestun*

Alexei Glutsyuk

Dynamical systems on torus modeling the dynamics of Josephson junction, isomonodromic deformations, and Painlevé 3 equation

Vladlen Timorin

The Mandelbrot set and its topological models

*This is a survey lecture on the Mandelbrot set focusing on its combinatorial structure. The Mandelbrot set $M$ describes dynamical bifurcations in a very specific 1-parameter family of holomorphic maps, namely, in the family of quadratic polynomials $z^2+c$. On the one hand, the chosen family is the “simplest possible” among non-trivial analytic families of analytic maps. On the other hand, it is so complicated as to include the subtlest features of almost any other non-trivial analytic family (the Mandelbrot set is “universal” by a result of C.McMullen). From a geometric viewpoint, $M$ is a complicated self-similar shape. However, it admits a simple combinatorial model $M_{comb}$ (due to Douady and Thurston), and there is a continuous monotone projection from $M$ to $M_{comb}$. The MLC conjecture (stating that $M$ is locally connected) would imply that this projection is a homeomorphism. We consider several approaches to defining $M_{comb}$ and some rougher models of $M$*

Sergey Cherkis

Instantons and bow varieties, II

*Bow and their associated structures form one step in a pattern generalizing quivers: Quivers, Bows, Slings, and Monowalls. Each of these, in turn, relates to
(1) geometry of gravitational instantons, (2) dynamics of gauge theory solitons, and (3) vacuum structure of supersymmetric gauge theories. After reviewing these connections and the string theory picture from which they emerge, we shall focus on bows and the precise mathematical construction of monopoles and instantons that they encode*

Andrey Gelash

Spontaneous modulation instability and exact multi-soliton solutions of the nonlinear Schrodinger equation

*I will talk about the modulation instability of a plane wave perturbed by noise in the framework of the integrable one-dimensional focusing nonlinear Schrodinger equation model.
Numerical simulations reveal intriguing statistical properties of the nonlinear stage of the modulation instability development, such as the formation of the Rayleigh probability density function for wave amplitudes.
We can explain the observed properties by employing specifically constructed exact multi-soliton solutions.
This approach poses several questions demanding further rigorous mathematical treatment*

Sergei Lukyanov

Density of states for the (Euclidean) black hole NLSM

*In Quantum Mechanics, it is often possible to reduce the computation of the density of states for the continuous component of the energy spectrum to that of finding the phase shifts which occur at the turning points of the classical trajectories. Such a simple-minded approach is not applicable for a general Quantum Field Theory, where the problem is not one-particle reducible. Nevertheless it turns out that the method works for some interesting situations in Integrable QFT. I will illustrate this for the case of the Non-Linear Sigma Model, whose target space can be interpreted as a two-dimensional black hole*

Sergei Korotkikh

Vertex models and (non)symmetric functions

*I plan to talk about connections between solvable vertex models and special functions arising from representation theory (mostly focusing on generalizations of Schur functions). It turns out that such functions can be expressed using partition functions of solvable vertex models, and algebraic properties of such objects can be inferred from the Yang-Baxter equation. I will start from the relation between the six-vertex model and Schur symmetric functions, and will continue by describing another known examples and related methods*

Sergey Cherkis

Instantons and bow varieties, I

*Bow and their associated structures form one step in a pattern generalizing quivers: Quivers, Bows, Slings, and Monowalls. Each of these, in turn, relates to
(1) geometry of gravitational instantons, (2) dynamics of gauge theory solitons, and (3) vacuum structure of supersymmetric gauge theories. After reviewing these connections and the string theory picture from which they emerge, we shall focus on bows and the precise mathematical construction of monopoles and instantons that they encode*

Andrei Smilga

Group manifolds and homogeneous spaces with HKT geometry: the role of automorphisms

*We present a new simple proof of the fact that certain group manifolds as well as certain homogeneous spaces $G/H$ of dimension $4n$ admit a triple of integrable complex structures that satisfy the quaternionic algebra and are covariantly constant with respect to the same torsionful Bismut connection, i.e. exhibit the HKT geometry. The key observation is that different complex structures are interrelated by automorphisms of the Lie algebra. To construct the quaternion triples, one only needs to construct the proper automorphisms, which is a more simple problem*

January 25, 2021

Alexander Bobenko

On a discretization of confocal quadrics: Geometric parametrizations, integrable systems and incircular nets

*We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system may be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. The theory is illustrated with a variety of examples in two and three dimensions. These include confocal coordinate systems parametrized in terms of Jacobi elliptic functions. Connections with incircular nets and elliptic billiards are established*

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