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Колесников Александр Викторович

Факультет математики

Профиль на hse.ru ↗ тел.: +7 (495) 772-95-90 | 15337
Публикаций
89
Языков
2
Наград
6
Конференций
5
Профиль Публикации (89) Курсы (13)

Профессиональные интересы

Теория оптимальной транспортировкиуравнения Монже-АмпераСоболевские пространстваизопериметрическое неравенствобесконечномерный анализВыпуклая геометриягеометрические потокиэллиптические и параболические уравнения в частных производныханализ на римановых многообразияхГауссовы мерыстохастика

Должности

  • Заместитель декана по учебной работе — Факультет математики
  • профессор — Факультет математики
  • Ведущий научный сотрудник — Факультет компьютерных наук, Институт искусственного интеллекта и цифровых наук, Международная лаборатория стохастических алгоритмов и анализа многомерных данных

Био

  • · Начал работать в НИУ ВШЭ в 2009 году.
  • · Научно-педагогический стаж: 19 лет.

Образование

  • 2006 · Доктор физико-математических наук: Математический институт им. В.А. Стеклова РАН, специальность 01.01.00 «Математика»
  • 2003 · Кандидат наук: специальность 01.01.00 «Математика»
  • 2002 · Аспирантура: Московский государственный университет им. М.В. Ломоносова, факультет: механико-математический
  • 1999 · Специалитет: Московский государственный университет им. М.В. Ломоносова, факультет: механико-математический, специальность «Математика, прикладная математика», квалификация «Математик»

Опыт работы

  • · 2009: Работает в НИУ ВШЭ с года

Награды и поощрения

  • · Медаль "Признание - 10 лет успешной работы" НИУ ВШЭ (май 2022)
  • · Надбавка за публикации, вносящие особый вклад в международную научную репутацию НИУ ВШЭ (2024–2026, 2022–2025)
  • · Надбавка за публикацию в международном рецензируемом научном издании (2020–2021, 2018–2020)
  • · Надбавка за статью в зарубежном рецензируемом журнале (2014–2016, 2012–2014)
  • · Надбавка за статью в зарубежном рецензируемом научном издании (2016–2018)
  • · Лучший преподаватель — 2021, 2012

Гранты и проекты

  • — · на соискание учёной степени кандидата наук

Конференции (5)

Показать все
  • · 2018: Коллоквиум "Москва-Пиза" (Москва). Доклад: Logarithmic Minkowski problem and optimal transportation
  • · 2018: Коллоквиум "Москва-Пиза" (Москва). Доклад: Logarithmic Minkowski problem and optimal transportation
  • · 2017: Probability and Analysis 2017 (Będlewo). Доклад: On KLS conjecture for certain classes of convex sets
  • · 2016: Stochastic Partial Differential Equations and Related Fields (Bielefeld). Доклад: Sobolev estimates for mass transportation mappings with application to transport equations and spectral gap
  • · 2015: Algebraic structures in convex geometry (Москва). Доклад: On the Monge-Ampere equation, related metric-measure spaces, and isoperimetric inequalities for convex bodies

Идентификаторы исследователя

Публикации (89)

Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities.

2015 · PREPRINT · en

Given a probability measure \mu supported on a convex subset \Omega of Euclidean space (\mathbb{R}^d,g_0), we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on (\Omega,g_0,\mu). To this end, we change the metric g_0 to a more general Riemannian one g, adapted in a certain sense to \mu, and perform our analysis on (\Omega,g,\mu). The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when \mu is unconditional, i.e. invariant under reflection with respect to the principle hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on (\Omega,g,\mu) tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--\'Emery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on(\Omega,g_0,\mu): refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincar\'e and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--\'Emery generalized Ricci curvature tensor, and the convexity of the manifold(\Omega,g,\mu). In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

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Exchangeable optimal transportation and log-concavity

2015 · PREPRINT · en

We study the Monge and Kantorovich transportation problems on R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of random variables is i.i.d.

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Remarks on mass transportation minimizing expectation of a minimum of affine functions

2015 · PREPRINT · en

We study Monge-Kantorovich problem with one-dimensional marginals μ,ν and the cost function c=min{l1,…,ln} which equals to minimum of a finite number n of affine functions li satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of n products Ii×Ji, where {Ii}, {Ji} are partitions of the line into unions of disjoint connected sets. The families of sets {Ii},{Ji} admit the following properties: 1) c=li on Ii×Ji, 2) {Ii},{Ji} is a couple of partitions solving an auxiliary n-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.

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Exchangeable optimal transportation and log-concavity

2015 · ARTICLE · en

We study the Monge and Kantorovich transportation problems on R∞R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on a Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of random variables is i.i.d.

Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures

2014 · ARTICLE · en

We study the optimal transportation mapping VΦ: ℝd → ℝd pushing forward a probability measure μ = e -V dx onto another probability measure ν = e-W dx. Following a classical approach of E. Calabi we introduce the Riemannian metric g = D2 Φ on ℝd and study spectral properties of the metric-measure space M = (ℝd, g,μ). We prove, in particular, that M admits a non-negative Bakry-Emery tensor provided both V and W are convex. If the target measure ν is the Lebesgue measure on a convex set Ω and μ is log-concave we prove that M is a CD(K, N) space. Applications of these results include some global dimension-free a priori estimates of \\D2 Φ||. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for M.

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Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary

2014 · PREPRINT · en

It is well known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension condition (combining the Ricci curvature with the "curvature" of the density). When the manifold has a boundary, the Reilly formula and its generalizations may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Poincar\'e-type inequalities on the manifold and on its boundary. For instance, we may handle Neumann conditions on a mean-convex domain, and obtain generalizations to the weighted-manifold setting of a purely Euclidean inequality of Colesanti, yielding a Brunn-Minkowski concavity result for geodesic extensions of convex domains in the manifold setting. All other previously known Poincar\'e-type inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux, Nguyen and Veysseire are recovered, in some cases improved, and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Finally, a new geometric evolution equation is proposed which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion previously confined to the linear setting, and for which a novel Brunn-Minkowski inequality in the weighted-Riemannian setting is obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative "dimension".

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Regularity of the Monge-Ampère equation in Besov's space

2014 · ARTICLE · en

Let \mu = e^{-V} \ dx be a probability measure and T = \nabla \Phi be the optimal transportation mapping pushing forward \mu onto a log-concave compactly supported measure \nu = e^{-W} \ dx. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi)} in the Besov spaces W^{\gamma,1}_{loc}. We prove that D^2 \Phi \in W^{\gamma,1}_{loc} provided e^{-V} belongs to a proper Besov class and W is convex. In particular, D^2 \Phi \in L^p_{loc} for some p>1. Our proof does not rely on the previously known regularity results.

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On continuity equations in infinite dimensions with non-Gaussian reference measure

2014 · ARTICLE · en

Let γ be a Gaussian measure on a locally convex space X and H be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order transportational PDE on X admits a weak solution under broad assumptions. Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures ν on $\R^{\infty}$, under the main assumption of integrability of logarithmic derivativesof v. We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures. measures.

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Remarks on the KLS conjecture and Hardy-type inequalities

2014 · ARTICLE · en

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body Ω ⊂ R n , not necessarily vanishing on the boundary ∂Ω. This reduces the study of the Neumann Poincar´e constant on Ω to that of the cone and Lebesgue measures on ∂Ω; these may be bounded via the curvature of ∂Ω. A second reduction is obtained to the class of harmonic functions on Ω. We also study the relation between the Poincar´e constant of a log-concave measure µ and its associated K. Ball body Kµ. In particular, we obtain a simple proof of a conjecture of Kannan–Lov´asz–Simonovits for unit-balls of ℓ n p , originally due to Sodin and Lata la–Wojtaszczyk.

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Weak regularity of Gauss mass transport

2014 · ARTICLE · en

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν , 2) T has the form T=ϕ∇ϕ|∇ϕ| , where ϕ is a function with convex sublevel sets. We prove a change of variables formula for T . We also establish Sobolev estimates for ϕ , and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDE's.

DOI ↗ PDF ↗

Курсы (13)