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Дунин-Барковский Петр Игоревич

Факультет математики

Профиль на hse.ru ↗ тел.: +7 (495) 772-95-90 | 12742
Публикаций
37
Языков
2
Наград
2
Конференций
13
Профиль Публикации (37) Курсы (12)

Профессиональные интересы

алгебраическая геометрияматематическая физикаинварианты узловкомбинаторика

Должности

  • Доцент — Факультет математики
  • Ведущий научный сотрудник — Факультет математики, Международная лаборатория кластерной геометрии
  • Заместитель заведующего лабораторией — Факультет математики, Международная лаборатория кластерной геометрии

Био

  • · Начал работать в НИУ ВШЭ в 2014 году.
  • · Научно-педагогический стаж: 11 лет.

Образование

  • 2015 · PhD: Амстердамский университет
  • 2014 · Кандидат физико-математических наук: Национальный исследовательский центр "Курчатовский институт"
  • 2011 · Магистратура: Московский физико-технический институт, специальность «Прикладные математика и физика», квалификация «Магистр прикладной математики и физики»
  • 2009 · Бакалавриат: Московский физико-технический институт, специальность «Прикладные математика и физика», квалификация «Бакалавр прикладных математики и физики»
  • — · Образование:Московский физико-технический институт, Москва, 2011. PhD

Опыт работы

  • · 2014: Работает в НИУ ВШЭ с года
  • · 2025: Проведение научных исследований по проекту «Геометрия и комбинаторика интегрируемых систем» (код проекта ФИ-2025-62) в лаборатории согласно техническому заданию под руководством С.К. Ландо год

Награды и поощрения

  • · Лучший преподаватель — 2019, 2017
  • · Группа высокого профессионального потенциала (кадровый резерв НИУ ВШЭ)Категория "Новые исследователи" (2015–2016)

Гранты и проекты

  • 2022 · 2020-2022 грант РНФ № 20-61-46005 «Алгебра и геометрия интегрируемых статистических моделей в старших размерностях» (участник проекта)
  • 2020 · 2019-2020 грант РНФ № 16-11-10316-П «Характеристические классы и теория представлений» (участник проекта)

Конференции (13)

Показать все
  • · 2024: Международная конференция "Инвариантность и интегрируемость 2" (г.Пушкин, Санкт-Петербург). Доклад: Symplectic duality and logarithmic topological recursion
  • · 2023: Инвариантность и интегрируемость (Репино). Доклад: Симплектическая двойственность в топологической рекурсии
  • · 2022: George Shabat 70 Conference (Москва). Доклад: Double Hurwitz numbers, maps, topological recursion, and dualities
  • · 2021: 7th Workshop on Combinatorics of Moduli Spaces, Cluster Algebras and Topological Recursion (Москва). Доклад: Explicit closed algebraic formulas for hypergeometric KP/generalized Hurwitz n-point functions
  • · 2019: Dynamics, Equations and Applications (DEA 2019) (Краков). Доклад: Loop equations and a proof of Zvonkine's qr-ELSV formula
  • · 2018: Workshop and School «Topological Field Theories, String theory and Matrix Models – 2018» (Москва). Доклад: On cut-and-join equation for monotone Hurwitz numbers
  • · 2017: Quantum information and topological recursion (Москва). Доклад: Proving topological recursion combinatorially for Hurwitz-type problems
  • · 2016: 5th Workshop on Combinatorics of Moduli Spaces, Hurwitz numbers, and cohomological field theories (Москва). Доклад: Dubrovin's superpotential as a global spectral curve
  • · 2016: 2016 American Mathematical Society von Neumann Symposium (Шарлотт, Северная Каролина). Доклад: Topological Recursion and Givental's Formalism: Spectral Curves for Gromov-Witten Theories
  • · 2015: Geometric Invariants and Spectral Curves (Leiden). Доклад: Superpotential as a global spectral curve
  • · 2015: Workshop and School "Quantum Geometry, Duality and Matrix Models" (Москва). Доклад: Dubrovin's superpotential as a global spectral curve
  • · 2014: "Group Theory and Knots" (Натал). Доклад: "Bi-colored and 4-colored maps, chains of matrices and spectral curve topological recursion".
  • · 2014: Symposium about Mathematics (Цюрих). Доклад: "Gromov-Witten theory and spectral curve topological recursion"

Идентификаторы исследователя

Публикации (37)

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.

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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 ↗

RNS superstring measure for genus 3

2025 · ARTICLE · en

We propose a new formula for the RNS superstring measure for genus 3. Our derivation is based on invariant theory. We follow Witten’s idea of using an algebraic parametrization of the moduli space (which he applied to re-derive D’Hoker and Phong’s formula for the RNS superstring measure for genus 2); but the particular parametrization that we use has not been applied to superstring theory before. We prove that the superstring measure is a linear combination (with complex coefficients) of three known functions. Furthermore, we conjecture the values of the coefficients of this linear combination and provide evidence for this conjecture. Unlike the Ansatz of Cacciatori, Dalla Piazza and van Geemen from 2008, our formula has a polar singularity along the hyperelliptic locus; the existence of this singularity was established by Witten in 2015. Moreover, our formula is not an Ansatz but follows from first principles, except for the values of the three coefficients

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.

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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 (12)