Lecture 01: An Introduction to Quantum Marginal Problems

Abstract: Classical multi-marginal problems ask whether a collection of lower-order marginals can arise from a common probability measure, and variational problems under such marginal constraints play an important role in optimal transport and density functional theory. Their quantum analogue is the $m$-representability, or quantum marginal, problem : determine whether prescribed reduced density operators arise as the marginals of a common $m$-particle quantum state. In particular, the fermionic $m$-representability problem–characterizing which reduced density matrices arise from an $m$-electron quantum state–remains a major outstanding problem in electronic structure theory and quantum chemistry. The goal of this introductory talk is to formulate these problems and describe some of the most accessible open questions.

Lecture 02: From Classical Relaxation to Noncommutative Obstructions: An Introductory Overview

Abstract: A basic difficulty in the search for minimizers over subsets of $W^{1,p}(\Omega;\mathbb{R}^D)$ is that weakly convergent sequences may develop increasingly fine oscillations that affect nonlinear energies of the form $I[u] = \int_{\Omega} F(x, u(x), \nabla u(x)) dx$.

In this introductory talk, I will review classical ideas originating in the work of Charles Morrey and explain, at a conceptual level, how quasiconvexification and gradient Young measures capture such oscillations and lead to relaxed variational problems for which existence can be recovered. I will then turn to variational problems involving noncommuting variables, where a different obstruction may occur : a minimizing sequence may converge at the level of noncommutative laws while the limiting law cannot be realized in the prescribed ambient algebra. I will illustrate this phenomenon through elementary examples related to noncommutative optimal transport and discuss a corresponding notion of relaxation obtained by enlarging the ambient algebra.

Lecture 03: Optimal transport and Mean Field Games on graphs

Abstract: Optimal transport on a finite graph provides a discrete analogue of Wasserstein geometry in which probability distributions are points of a finite-dimensional simplex. In this introductory talk, I will explain how this geometry leads naturally to Hamilton–Jacobi equations, mean field game systems, and master equations on the probability simplex. A central issue is that the boundary of the simplex corresponds to vanishing probabilities, where the underlying transport geometry may become singular. I will describe a quantitative preservation-of-positivity estimate which prevents solutions from reaching the boundary in finite time and leads to a classical well-posedness theory for the master equation on the open simplex without imposing boundary conditions. I will also discuss how the associated continuous-time Markov-chain game can be recovered from the PDE structure and how this finite-state theory may provide a structure-preserving approximation of continuous-state mean field games.

Biography

Wilfrid Gangbo is a Distinguished Research Professor at the University of California, Los Angeles (UCLA), in the Mathematics Department. He was born in Benin, West Africa and completed his K12 education there before moving to Switzerland where he received his undergraduate education. Gangbo received his PhD in 1992 at EPFL, Switzerland (Swiss Federal Institute of Technology) under Bernard Dacorogna. He joined the faculty at the Georgia Institute of Technology (Georgia Tech) where he rose to the rank of professor of mathematics in 2002. In 2016 he became a professor of mathematics at UCLA.

In 2022, Gangbo was elected a Fellow of the American Academy of Arts and Sciences and received the 2026 AMS Award for Distinguished Public Service. He has delivered many distinguished lectures at national and international meetings such as plenary addresses at JMM 2023 and 2025, the 2021 Richard M. Karp Distinguished Lecture at the Simons Institute in Berkeley and the Claytor–Woodard Address at JMM 2017. He was a visiting chancellor’s professor at UC Berkeley in 2018–2019, CNRS director of research on visiting positions at ENS-Paris, ENS-Lyon and visiting professor at Université Paris–Dauphine. Gangbo served as a program director at the division of mathematical sciences of the National Science Foundation from 2012 to 2013. He chaired the AMS Committee on the Profession from 2019 to 2021 and currently serves on many national and international committees. He serves on the editorial boards of numerous journals and remains involved in scientific activities in Africa, especially in Senegal and Benin, his home country.