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Казарян Максим Эдуардович

Факультет математики

Профиль на hse.ru ↗ тел.: +7 (495) 772-95-90 | 12744
Публикаций
38
Языков
1
Наград
7
Конференций
5
Профиль Публикации (38) Курсы (9)

Профессиональные интересы

топологиягеометриятеория особенностейхарактеристические классыпространства модулейперечислительные задачи комплексной геометриистратификация пространств модулей кривых и мероморфных функций

Должности

  • Профессор — Факультет математики, Базовая кафедра Математического института им. В.А. Стеклова РАН
  • Научный руководитель — Факультет математики, Международная лаборатория кластерной геометрии

Био

  • · Начал работать в НИУ ВШЭ в 2008 году.
  • · Научно-педагогический стаж: 35 лет.

Образование

  • 2003 · Доктор физико-математических наук
  • 1992 · Кандидат физико-математических наук
  • 1988 · Специалитет: Московский авиационный институт им. С. Орджоникидзе, специальность «Прикладная математика», квалификация «Инженер-математик»

Опыт работы

  • · 2021-н.в.: ведущий научный сотрудник, международная лаборатория кластерной геометрии, НИУ ВШЭ

Награды и поощрения

  • · Благодарность факультета математики НИУ ВШЭ (декабрь 2025)
  • · Благодарность НИУ ВШЭ (март 2025)
  • · Надбавка за академическую работу (2025–2026, 2021–2022)
  • · Надбавка за публикацию в журнале из Списка А (и приравненном к нему научном издании) (2024–2025, 2023–2024)
  • · Надбавка за публикацию в международном рецензируемом научном издании (2022–2023, 2018–2020)
  • · Надбавка за статью в зарубежном рецензируемом научном издании (2016–2018)
  • · Лучший преподаватель — 2025

Гранты и проекты

  • 2026 · 2024-2026 Грант РНФ №24-11-00366 по теме «Теория особенностей и интегрируемость» (Научный руководитель)
  • 2020 · 2019-2020 Грант РНФ № 16-11-10316-П по теме «Характеристические классы и теория представлений» (участник проекта)
  • — · 2013 Грант РФФИ 13-01-00383\14 по теме "Комбинаторные и топологические методы исследования функциональных пространств" (участник проекта)

Конференции (5)

Показать все
  • · 2023: научная работа в рамках проекта "Кластерные алгебры и пространства модулей плоских и голоморфных связностей" (Хайфа). Доклад: Топологическая рекурсия и решения интегрируемых систем уравнений математической физики
  • · 2016: 5th Workshop on Combinatorics of Moduli Spaces, Hurwitz numbers, and cohomological field theories (Москва). Доклад: Universal cohomological expressions for strata in genus zero Hurwitz spaces
  • · 2014: The Legacy of Vladimir Arnold (Торонто). Доклад: On Salmon's enumeration of tangential singularities
  • · 2014: 4th Workshop on Combinatorics of Moduli Spaces, Cluster Algebras, and Topological Recursion (Москва (Moscow)). Доклад: Enumeration of Grotendieck's dessins 2: Topological recursion
  • · 2013: Integrable Systems an Moduli Spaces (

Идентификаторы исследователя

Публикации (38)

The universal gl-weight system and the chromatic polynomial

2026 · ARTICLE · en

Weight systems associated to the Lie algebras 𝔤𝔩(N) for N = 1,2,... can be unified into auniversal one. The construction is based on an extension of the 𝔤𝔩(N) weight systems to permutations. This universal weight system takes values in the algebra of polynomials C[N;C1,C2,...] in infinitely many variables. We show that under the substitution Cm = xNm−1, m = 1,2,..., the leading term in N of the value of the universal 𝔤𝔩 weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. Moreover, we show that under the substitution Cm = pmNm−1, m = 1,2,..., the leading term in N of the value of the universal 𝔤𝔩-weight system determines a filtered Hopf algebra homomorphism from the rotational Hopf algebra of permutations, which we construct in the present paper, to the Hopf algebra of polynomials C[p1,p2,...].

DOI ↗ PDF ↗

Blobbed topological recursion and KP integrability

2026 · ARTICLE · en

We revise the notion of the blobbed topological recursion by extending it to the setting of generalized topological recursion as well as allowing blobs which do not necessarily admit topological expansion. We show that the so-called non-perturbative differentials form a special case of this revisited version of blobbed topological recursion. Furthermore, we prove the KP integrability of the differentials of blobbed topological recursion for the input data that include KP-integrable blobs. This result generalizes, unifies, and gives a new proof of the KP integrability of nonperturbative differentials conjectured by Borot–Eynard and recently proved by the authors.

DOI ↗ PDF ↗

Symplectic duality for topological recursion

2025 · ARTICLE · en

We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal parameters (up to some maximal length on both sides), and on one side there are also distinguished cycles controlled by degrees of formal variables. In these variables the weighted double Hurwitz numbers are presented as coefficients of expansions of some differentials that we prove to satisfy topological recursion. Our results partly resolve a conjecture that we made in [Comm. Math. Phys. 402 (2023), pp. 665–694] and are based on a system of new explicit functional relations for the more general (m,n)-correlation functions, which correspond to the case when there are distinguished cycles controlled by formal variables in both special singular fibers. These (m,n)-correlation functions are the main theme of this paper and the latter explicit functional relations are of independent interest for combinatorics of weighted double Hurwitz numbers. We also put our results in the context of what we call the “symplectic duality”, which is a generalization of the x−y duality, a phenomenon known in the theory of topological recursion.

DOI ↗ PDF ↗

A universal formula for the x−y swap in topological recursion

2025 · ARTICLE · en

We prove a recent conjecture of Borot et al. that a particular universal closed algebraic formula recovers the correlation differentials of topological recursion after the swap of x and y in the input data. We also show that this universal formula can be drastically simplified (as it was already done by Hock). As an application of this general x-y swap result, we prove an explicit closed formula for the topological recursion differentials for the case of any spectral curve with unramified y and arbitrary rational x.

DOI ↗ PDF ↗

Degenerate and Irregular Topological Recursion

2025 · ARTICLE · en

We use the theory of x-y duality to propose a new definition/construction for the correlation differentials of topological recursion; we call it generalized topological recursion. This new definition coincides with the original topological recursion of Chekhov–Eynard–Orantin in the regular case and allows, in particular, to get meaningful answers in a variety of irregular and degenerate situations.

DOI ↗ PDF ↗

KP integrability through the x-y swap relation

2025 · ARTICLE · en

We discuss a universal relation that we call the x-y swap relation, which plays a prominent role in the theory of topological recursion, Hurwitz theory, and free probability theory. We describe in a very precise and detailed way the interaction of the x-y swap relation and KP integrability. As an application, we prove a recent conjecture that relates some particular instances of topological recursion to the Mironov–Morozov–Semenoff matrix integrals.

DOI ↗ PDF ↗

Generalized chord diagrams and weight systems

2025 · ARTICLE · en

The paper is devoted to a description of the recent progress in understanding the extension of Lie algebra weight systems to permutations. Lie algebra weight systems are functions on chord diagrams arising naturally in Vassiliev's theory of finite-type knot invariants. These functions satisfy certain linear restrictions known as Vassiliev's 4-term relations. Chord diagrams can be interpreted as fixed-point-free involutions in symmetric groups, and an extension of Lie algebra weight systems to arbitrary permutations was aimed at finding an efficient way to compute their values. We show that this extension is of interest on its own, which suggests introducing the notion of weight system on permutations. To this end we define generalized Vassiliev's relations for permutations, which reduce to conventional ones for chord diagrams. We also describe the corresponding Hopf algebra structures on spaces of permutations that match the classical Hopf algebra structure on the space of chord diagrams modulo 4-term relations. Among main results of the paper is an explicit formula for the average value of the universal gl-weight system on permutations. This formula implies, in particular, that this average value is a tau-function for the Kadomtsev–Petviashvili hierarchy of partial differential equations. Its proof is based on an analysis of a quantum version of the universal gl-weight system. Bibliography: 33 titles.

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Log Topological Recursion Through the Prism of x-y Swap

2024 · ARTICLE · en

We introduce a new concept of logarithmic topological recursion that provides a patch to topological recursion in the presence of logarithmic singularities and prove that this new definition satisfies the universal x-y swap relation. This result provides a vast generalization and a proof of a very recent conjecture of Hock. It also uniformly explains (and conceptually rectifies) an approach to the formulas for the -point functions proposed by Hock.

DOI ↗ PDF ↗

Symplectic duality via log topological recursion

2024 · ARTICLE · en

We review the notion of symplectic duality earlier introduced in the context of topological recursion. We show that the transformation of symplectic duality can be expressed as a composition of x-y dualities in a broader context of log topological recursion. As a corollary, we establish nice properties of symplectic duality: various convenient explicit formulas, invertibility, group property, compatibility with topological recursion and KP integrability. As an application of these properties, we get a new and uniform proof of topological recursion for large families of weighted double Hurwitz numbers; this encompasses and significantly extends all previously known results on this matter.

DOI ↗ PDF ↗

Polynomial Relations Among Kappa Classes on the Moduli Space of Curves

2024 · ARTICLE · en

We construct an infinite collection of universal—independent of (g,n)—polynomials in the Miller–Morita–Mumford classes κm ∈ H2m(Mg,n, Q), defined over the moduli space of genus g stable curves with n labeled points. We conjecture vanishing of these polynomials in a range depending on g and n.

DOI ↗ PDF ↗

Курсы (9)