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Забродин Антон Владимирович

Факультет математики

Профиль на hse.ru ↗ тел.: 12728
Публикаций
78
Языков
1
Наград
4
Конференций
8
Профиль Публикации (78) Курсы (6)

Профессиональные интересы

Интегрируемые системы классической и квантовой физикиТеория случайных матриц и ее приложенияконформная теория поляМатематические модели процессов роста и формирования структур

Должности

  • профессор — Факультет математики
  • Старший научный сотрудник — Факультет математики

Био

  • · Начал работать в НИУ ВШЭ в 2011 году.
  • · Научно-педагогический стаж: 42 года.

Образование

  • 1999 · Доктор физико-математических наук
  • 1989 · Кандидат физико-математических наук
  • 1984 · Специалитет: Московский государственный университет им. М.В. Ломоносова, специальность «Физика», квалификация «Физик»

Опыт работы

  • · 2011: Работает в НИУ ВШЭ с года

Награды и поощрения

  • · Надбавка за публикацию в журнале из Списка А (и приравненном к нему научном издании) (2025–2026, 2024–2025, 2023–2024)
  • · Надбавка за публикацию в международном рецензируемом научном издании (2022–2023, 2021–2022, 2019–2021, 2017–2019)
  • · Надбавка за статью в зарубежном рецензируемом журнале (2015–2017, 2013–2015)
  • · Лучший преподаватель — 2012

Гранты и проекты

  • — · 2014 Грант РФФИ 14-01-90405 "Теория представлений, гомологическая алгебра и интегрируемые системы, российско-украинский грант (участник проека)

Конференции (8)

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  • · 2025: V Конференция математических центров России (Красноярск). Доклад: Интегрируемая деформация системы Руйсенарса-Шнайдера
  • · 2021: Международная конференция «Интегрируемость», посвященная 75-летию А. К. Погребкова (Москва). Доклад: Field analog of the Ruijsenaars-Schneider model from elliptic families of solutions to the 2D Toda lattice
  • · 2018: Классические и квантовые интегрируемые системы (Протвино). Доклад: Спиновое обобщение иерархии Калоджеро-Мозера и матричная иерархия КП
  • · 2016: Classical and quantum integrable systems and supersymmetry (Тянцзинь). Доклад: Supersymmetric quantum spin chains and classical integrable models
  • · 2015: LPT ENS (Париж). Доклад: «Квантовые спиновые цепочки и интегрируемые системы классической механики»
  • · 2015: семинар Кобе по интегрируемым системам (Кобе). Доклад: «Спектры квантовых магнетиков и интегрируемые системы многих частиц классической механики»
  • · 2014: Международная конференция "Физика и математика нелинейных явлений" (Галлиполи). Доклад: "Бездисперсионная иерархия DKP и эллиптическое уравнение Левнера"
  • · 2013: Synthesis of integrabilities in the context of duality between the string theory and gauge theories (Москва). Доклад: Quantum transfer matrices and classical tau-functions

Идентификаторы исследователя

Публикации (78)

Elliptic Solutions of the Toda Lattice with Constraint of Type B and Deformed Ruijsenaars–Schneider System

2023 · ARTICLE · en

We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars-Schneider system. We find its commutation representation in the form of the Manakov triple and study properties of the spectral curve. By studying more general elliptic solutions (elliptic families), we also suggest an extension of the deformed Ruijsenaars-Schneider system to a field theory.

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Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation

2022 в печати · PREPRINT · en

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov-Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the L ̈owner type which we call the symmetric radial L ̈owner equation. It is also shown that solutions to the symmetric radial L ̈owner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle.

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Constrained Toda hierarchy and turning points of the Ruijsenaars–Schneider model

2022 · ARTICLE · en

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint L¯=L†L¯=L† on the two Lax operators (in the symmetric gauge). We prove the existence of the tau function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau function. In this and some other respects, the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars–Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchanges the marked points in which the Baker–Akhiezer function has essential singularities.

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Dyson gas on a curved contour

2022 · ARTICLE · en

We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson's gas (the β-ensemble) on the unit circle. We compute the non-vanishing terms of the large N expansion of the free energy (N is the number of particles) by iterating the 'loop equation' that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann–Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour.

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Field analogue of the Ruijsenaars-Schneider model

2022 · ARTICLE · en

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of L-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing η, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit η → 0 is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed.

DOI ↗ PDF ↗

Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation

2022 в печати · ARTICLE · en

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov–Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the Löwner type which we call the symmetric radial Löwner equation. It is also shown that solutions to the symmetric radial Löwner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle.

DOI ↗ PDF ↗

Löwner equations and reductions of dispersionless hierarchies

2021 · ARTICLE · en

The equations of Löwner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Löwner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarchies (dKP, dBKP, dToda, and dDKP). The one-variable reductions are described by solutions of different versions of Löwner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP. We also discuss multi-variable reductions which are given by a system of Löwner equations supplemented by a system of partial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.

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Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

2021 · ARTICLE · en

We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the "zeroth" time t(0) = x. It is known that their poles as functions of t1 move as particles of the elliptic RuijsenaarsSchneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians that govern the dynamics of poles with respect to the mth hierarchical times t(m) and (t) over bar (m) of the 2D Toda lattice hierarchy are obtained from the expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points.

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Elliptic solutions to the KP hierarchy and elliptic Calogero–Moser model

2021 · ARTICLE · en

We consider solutions of the Kadomtsev-Petviashvili hierarchy which are elliptic functions of x = t (1). It is known that their poles as functions of t (2) move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian H ( k ) of the elliptic Calogero-Moser model which governs the dynamics of poles with respect to the kth hierarchical time. The Hamiltonians H ( k ) are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker-Akhiezer function has essential singularity.

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Kadomtsev–Petviashvili Turning Points and CKP Hierarchy

2021 · ARTICLE · en

A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. A new identity for theta-functions of curves with holomorphic involution having fixed points is obtained.

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Курсы (6)