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Бычков Борис Сергеевич

Факультет математики

Профиль на hse.ru ↗ тел.: +7 (495) 772-95-90 | 12728
Публикаций
25
Языков
1
Наград
5
Конференций
20
Профиль Публикации (26) Курсы (10)

Профессиональные интересы

числа Гурвицапространства модулей комплексных кривых

Должности

  • Доцент — Факультет математики

Био

  • · Начал работать в НИУ ВШЭ в 2011 году.
  • · Научно-педагогический стаж: 14 лет.

Образование

  • 2015 · Кандидат физико-математических наук
  • 2008 · Специалитет: Московский государственный университет им. М.В. Ломоносова, факультет: механико-математический, специальность «Математика», квалификация «Математик»
  • — · Аспирантура: Национальный исследовательский университет «Высшая школа экономики», факультет: математический

Награды и поощрения

  • · Надбавка за академические успехи и вклад в научную репутацию НИУ ВШЭ (2023, 2023)
  • · Надбавка за публикацию в журнале из Списка B (2023–2024)
  • · Надбавка за публикацию в международном рецензируемом научном издании (2021–2022, 2021–2022, 2020–2022)
  • · Лучший преподаватель — 2021–2022, 2018–2019, 2015–2016
  • · Группа высокого профессионального потенциала (кадровый резерв НИУ ВШЭ)Категория "Новые исследователи" (2016)

Гранты и проекты

  • — · на соискание учёной степени кандидата наук

Конференции (20)

Показать все
  • · 2021: 7th Workshop on Combinatorics of Moduli Spaces, Cluster Algebras and Topological Re-cursion (Москва). Доклад: Topological recursion for hypergeometric KP/generalized Hurwitz n-point functions
  • · 2021: IV международная конференция «Группы и квандлы в маломерной топологии» (Новосибирск). Доклад: «Топологическая рекурсия для карт и пополненных простых карт»
  • · 2021: Workshop on Classical and Quantum Integrable Systems (CQIS 2021) (Сочи). Доклад: Topological recursion for KP tau functions of hypergeometric type
  • · 2021: Конференция международных математических центров мирового уровня (Сочи). Доклад: Electrical networks and Lagrangian Grassmanians
  • · 2021: 3rd International Conference on Integrable Systems and Nonlinear Dynamics (Ярославль). Доклад: Electrical networks and Lagrangian Grassmanians
  • · 2020: 2nd International Conference on Integrable Systems and Nonlinear Dynamics (ISND 2020) (Ярославль). Доклад: Star-triangle transformation of the Potts model partition function as a solution for the tetrahedron equation and related combinatorial topics
  • · 2019: Mathematical Sciences Departmental Seminar (Колчестер). Доклад: Hurwitz numbers and integrable hierarchies
  • · 2018: Центр интегрируемых систем (Ярославль). Доклад: Алгебра для производящих рядов двойных чисел Гурвица
  • · 2018: Интегрируемые системы и нелинейная динамика (Ярославль). Доклад: Полиномиальные инварианты графов и линейные иерархии
  • · 2017: Quantum information and topological recursion (Москва). Доклад: Degrees of strata of Hurwitz spaces
  • · 2017: Summer School in Enumerative Geometry (Триест). Доклад: Degrees of strata of Hurwitz spaces
  • · 2017: Kezenoi-am 2017 (Грозный). Доклад: Quasi-polynomiality of Bousquet-Melou-Schaeffer numbers
  • · 2017: Several Complex Variables (Красноярск). Доклад: Degrees of strata of Hurwitz spaces
  • · 2016: Classical and quantum integrable systems and supersymmetry (Тянцзинь). Доклад: Degrees of the strata of Hurwitz spaces
  • · 2015: Young People week in singularities (Марсель). Доклад: On decomposition of the cyclic permutation into the product of a given number of permutations
  • · 2015: Пятая школа-конференция по алгебраической геометрии и комплексному анализу для молодых математиков России (Коряжма, Архангельская область). Доклад: Степени когомологических классов на стратах дискриминанта пространства Гурвица
  • · 2014: Primitive forms and related subjects (柏市 (Kashiwa)). Доклад: On the number of coverings of the sphere ramified over given points
  • · 2014: Graduate workshop on Moduli of curves (Stony Brook). Доклад: On the geomtery of decompositionof the cyclic permutation into the product of a given number of permutations
  • · 2013: Geometry days in Novosibirsk 2013 (Новосибирск). Доклад: The computation of megamaps
  • · 2010: International Workshop "Classical and quantum integrable systems and supersymmetry" (Тяньцзинь). Доклад: Degrees of the stratum of Hurwitz spaces

Идентификаторы исследователя

Публикации (26)

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 ↗

Symplectic geometry of electrical networks

2025 · ARTICLE · en

In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks in the disc obtained in \cite{L}, \cite{CGS} and \cite{BGKT}. In particular, we state an explicit connection between these works and describe some of the combinatorics developed there in the language of symplectic geometry. We also show that the combinatorics of the concordance vectors forces the uniqueness of the symplectic form, such that corresponding points of the Grassmannian are isotropic. We define a notion of Lagrangian concordance which provides a construction of the compactification of the space of electrical networks in the positive part of the Lagrangian Grassmannian bypassing the construction from \cite{L}

DOI ↗ PDF ↗

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 ↗

Topological recursion, symplectic duality, and generalized fully simple maps

2024 · ARTICLE · en

For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the n-point functions produced by the topological recursion on these curves via the n-point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions.

DOI ↗

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

2024 · ARTICLE · en

We study the n-point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ℏ2-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so-called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ℏ2-deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)-type formulas discussed in the literature

DOI ↗

Курсы (10)