Andrii Bobrov

M.Sc. Student in Mathematics, Kyiv Academic University (KAU)
Visiting Member, The Fields Institute for Research in Mathematical Sciences

Email: a.bobrov@kau.edu.ua  |  Curriculum Vitae (PDF)  |  0009-0005-9707-5619

About & Research Interests

I am a Master's student in Mathematics at Kyiv Academic University (joint program with the Institute of Mathematics of the NAS of Ukraine), advised by Georgii Riabov.

My research focuses on Functional Analysis, Optimal Transport Theory (Monge–Kantorovich and Beckmann formulations), Partial Differential Equations, and Infinite-Dimensional Probability.

Research & Manuscripts

• Variational Formulations of the Monge–Kantorovich Problem in Banach Spaces
  (with G. Riabov) — Manuscript in preparation. [ Working draft available upon request]
  Investigating minimal $L^1$-norm vector fields satisfying $-\operatorname{div}(u\mu) = \rho\mu$, using Fenchel–Rockafellar duality, and transport ray disintegration on separable Banach spaces.
• Optimal Transport on Graphs and the Beckmann Problem (2025)
  B.Sc. Thesis, KPI IPT (Advisor: G. Riabov). [PDF (in Ukrainian; includes English abstract)]

Selected Talks

• Variational Formulations of Optimal Transport in Banach Spaces
  Working Seminar on Optimal Transport, The Fields Institute, Toronto, Canada (Scheduled: Fall 2026).
• Kantorovich–Rubinstein Distance on Gaussian Spaces
  The Skorokhod Readings & Int. Conference of Young Mathematicians, Kyiv, 2026.
  [Slides (PDF)]
  [Abstract]

  We consider the Kantorovich–Rubinstein ($W_1$) optimal transport problem on Gaussian spaces equipped with a Cameron–Martin structure, formulated through the divergence constraint $Iu = g$, where $I = D^*$ is the extended stochastic integral (the adjoint of the Malliavin derivative).
  
  The talk covers three main aspects:

  • One-Dimensional Case: The divergence equation reduces to a first-order linear ODE with a Gaussian integrating factor, admitting an explicit, unique solution in terms of cumulative distribution functions that directly recovers the classical $W_1$ distance.
  • Finite-Dimensional Case ($\mathbb{R}^d$): We reformulate the setup into the continuous Beckmann minimal flow problem and demonstrate the loss of uniqueness for the divergence equation via rotational fields. Existence of a minimizing vector Radon measure is obtained through weak-* compactness and lower semicontinuity.
  • Infinite-Dimensional Setting (Banach Space): Addressing the lack of local compactness, we establish the existence of a minimizing $H$-valued transport measure by combining the tightness of marginals (via Ulam's and Mazur's theorems) with Prokhorov's theorem, and outline open problems regarding the absolute continuity and uniqueness of the resulting measure.
• Barycenters in Wasserstein Space: Local Geometry and Parametric Convergence
  KAU Mathematics Research Seminar, Kyiv, 2026.
  [Abstract]

  Fréchet barycenters in the Wasserstein space $\mathcal{W}_2(\mathbb{R}^d)$ provide a principled framework for interpolating and averaging probability distributions, with applications ranging from image processing to non-Euclidean statistics. In classical Hilbert spaces, the empirical Fréchet mean converges to the population mean at a dimension-free parametric rate $\mathbb{E}\|b_n - b^*\|^2 = \frac{1}{n}\operatorname{Var}(X)$. However, transferring such statistical guarantees to $\mathcal{W}_2(\mathbb{R}^d)$ is challenging due to the non-linear, metric-geodesic nature of Wasserstein spaces.
  
  In this talk, we investigate how the local metric geometry of $\mathcal{W}_2(\mathbb{R}^d)$ governs the convergence of empirical barycenters. We construct the space of directions and metric tangent cones via comparison angles and geodesics, highlighting that for absolutely continuous base measures $\mu \in \mathcal{P}_{ac}(\mathbb{R}^d)$, the tangent cone $T_\mu \mathcal{W}_2(\mathbb{R}^d)$ embeds as a closed convex cone in $L^2(\mu)$. Exploiting this local structure and displacement convexity, we present conditions ensuring the existence and uniqueness of barycenters for random measures on $\mathcal{W}_2(\mathbb{R}^d)$, and establish an explicit finite-sample parametric convergence bound: $$\mathbb{E}\big[W_2^2(b_n, b^*)\big] \le \frac{4\sigma^2}{C n},$$ where $\sigma^2$ is the barycentric variance and $C \in (0, 1]$ is a geometric parameter reflecting local curvature and strict convexity.

Education

• M.Sc. in Mathematics, Kyiv Academic University (2025 – Expected 2027)
• B.Sc. in Applied Mathematics, KPI IPT (2021 – 2025)