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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)

Poincaré and Brunn--Minkowski inequalities on the boundary of weighted Riemannian manifolds

2018 · ARTICLE · en

We study a Riemannian manifold equipped with a density which satisfies the Bakry--\'Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincar\'e-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obtained, yielding spectral-gap estimates for the weighted Laplacian on the boundary. Motivated by these inequalities, a new geometric evolution equation is proposed, which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion thus far confined to the purely linear setting. This geometric flow is characterized by having parallel normals (of varying velocity) to the evolving hypersurface along the trajectory, and is intimately related to a homogeneous Monge-Amp\`ere equation on the exterior of the convex domain. Using the aforementioned Poincar\'e-type inequality on the boundary of the evolving hypersurface, we obtain a novel Brunn--Minkowski inequality in the weighted-Riemannian setting, amounting to a certain concavity property for the weighted-volume of the evolving enclosed domain. All of these results appear to be new even in the classical non-weighted Riemannian setting.

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The KLS Isoperimetric Conjecture for Generalized Orlicz Balls

2018 · ARTICLE · en

What is the optimal way to cut a convex bounded domain K in Euclidean space (Rn,|⋅|) into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lov\'asz and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of n) in the surface area, one might as well dissect K using a hyperplane. This conjectured essential equivalence between the former non-linear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volumetric and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls K={x∈Rn;∑i=1nVi(xi)≤E}, confirming its validity for certain levels E∈R under a mild technical assumption on the growth of the convex functions Vi at infinity (without which we confirm the conjecture up to a log(1+n) factor). In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry is required from K. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.

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On multistochastic Monge-Kantorovich problem, bitwise operations, and fractals

2018 · PREPRINT · en

The multistochastic (n,k)-Monge--Kantorovich problem on a product space ∏ni=1Xi is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×…×Xik for all k-tuples {i1,…,ik}⊂{1,…,n} for a given 1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n=3,k=2,Xi=[0,1], the cost function c(x,y,z)=xyz, and the corresponding two--dimensional projections are Lebesgue measures on [0,1]2. We prove, in particular, that the mapping (x,y)→x⊕y, where ⊕ is the bitwise addition (xor- or Nim-addition) on [0,1]≅Z∞2, is the corresponding optimal transportation. In particular, the support of π is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.

On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type

2018 · PREPRINT · en

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure γ enjoys 1n-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L, γ(λK+(1−λ)L)12n≥λγ(K)12n+(1−λ)γ(L)12n. Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to p-concavity, with p>0, with respect to the addition of symmetric convex se

Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem

2018 · PREPRINT · en

We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩. We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−1 endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that K satisfies the following analog of the Gaussian transportation inequality for the uniform measure σ on Sn−1: 1nEnt(ν)≥K(σ,ν). We show that there exists a remarkable similarity between our results and the theory of the K{\"a}hler-Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.

Brascamp-Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

2017 · ARTICLE · en

It is known that by dualizing the Bochner–Lichnerowicz–Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula 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 Brascamp–Lieb-type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are 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 generalized dimension.

DOI ↗ PDF ↗

Remarks on curvature in the transportation metric

2017 · ARTICLE · en

According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kähler–Einstein equation eΦ = detD2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D2Φ, eΦdx) has a non-positive Bakry–Émery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Émery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.

DOI ↗ PDF ↗

Sharp Poincaré-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets

2017 · CHAPTER · en

A sharp Poincaré-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso-second-variation inequality. The new inequality is nothing but an infinitesimal equivalent form of Ehrhard’s inequality for the Gaussian measure. While Ehrhard’s inequality does not extend to general CD(1, ∞) measures, we formulate a sufficient condition for the validity of Ehrhard-type inequalities for general measures on RnRn via a certain property of an associated Neumann-to-Dirichlet operator.

DOI ↗

Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry.

2017 · ARTICLE · en

We consider probability measures on R∞ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. We show in the latter case that existence problem for optimal transportation is closely related to ergodicity of the target measure. In particular, we prove existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.

DOI ↗ PDF ↗

Extremal Kaehler-Einstein metric for two-dimensional convex bodies

2017 · PREPRINT · en

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{\"a}hler-Einstein equation $e^{-\Phi} = \det D^2 \Phi$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 \Phi$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 \Phi$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.

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Курсы (13)