Seminar on Analysis and Probability, organized by the HSE Faculty of Mathematics

Organizing committee:
     Cedric Bernardin  
     Andrey Dymov
     Alexander Kolesnikov
     Valentin Konakov  
     Sergey Kuksin 
     Alexander Menovschikov

The seminar is usually held online unless otherwise stated.

Video recordings of 2025

February 12, 2026   18:00, online

Sergio Polidoro . Taylor formulas for Nonlocal Kinetic Equations and Lévy Processes

Abstract:  

We consider spaces of Hölder continuous functions suitable for studying the regularity theory for non local kinetic operators acting on the phase space. We first recall the geometrical setting where the regularity theory of the analogous purely differential operators is usually studied. Then we state and discuss an intrinsic Taylor-like formula taylored on the relevant non-Euclidean geometry. We then examine the application of our results to some Lévy Processes.
These results have been proved in a joint work in collaboration with Maria Manfredini and Stefano Pagliarani. 

February 05, 2026   16:00, online

Tadahisa Funaki . Interface motion from non-gradient Glauber-Kawasaki dynamics

Abstract:  

Motivated by the problem of the dynamic phase transition, we discuss the derivation of interface motion, described by the anisotropic curvature flow, from Glauber-Kawasaki dynamics of non-gradient type. Our result is formulated as the hydrodynamic limit at the quantitative level, that is, we can derive the convergence rate. We apply the method recently developed in quantitative homogenization. We also present some results for the fluctuation of the interface. The talk is based on joint works with Chenlin Gu (Tsinghua), Han Wang (Tsinghua), Hyunjoon Park (Meiji), Claudio Landim (IMPA) and Sunder Sethuraman (Arizona).

January 29, 2026   18:00, online + room 108

Alexey Glutsyuk. On dynamical systems on the torus modeling the Josephson junction: Heun and Painlevé III equations

Abstract:  

In 1962 B. Josephson predicted a tunnelling effect arising in a system of two superconductors separated by a thin dielectric layer, now known as the Josephson junction. It involves the existence of a supercurrent crossing the junction and governed by equations discovered by Josephson.

The overdamped Josephson junction is modeled by a family of differential equations on the two-dimensional torus, known as the RSJ model, depending on three parameters: the abscissa B, the ordinate A, and a fixed frequency of external forcing. The corresponding rotation number is a function of (B,A). The phase-lock areas are those level sets of the rotation number that have non-empty interior; they exist only for integer rotation numbers. This quantization phenomenon was discovered by V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, who showed that the system is equivalent to a family of second-order linear differential equations on the Riemann sphere: special double confluent Heun equations.

The phase-lock areas form garlands of infinitely many domains separated by points. Those separation points that do not lie on the abscissa axis are called constrictions. In joint work with Yu. P. Bibilo, the speaker showed that in each phase-lock area all constrictions lie on a single vertical line whose abscissa equals the product of the forcing frequency and the rotation number. The proof uses the above equivalence with linear equations, Stokes phenomena, isomonodromic deformations governed by Painlevé III equations, holomorphic vector bundles in the spirit of A. A. Bolibruch’s work, and methods from slow–fast system theory. Similar methods were recently used by the speaker to compute the genus of spectral curves associated with special double confluent Heun equations admitting polynomial solutions.

A. S. Gorsky asked whether there exist dynamical systems on the torus equivalent to general Heun equations with four singularities. A family of such systems, obtained as a deformation of the RSJ model, was recently constructed by the speaker. In this deformed model all constrictions disappear. This is joint work with A. A. Alexandrov.

In the talk, we sketch proofs of the above results and give a survey of related developments and open questions.