Description

The optimal transport problem was first introduced by Gaspard Monge in 1781, who was probably motivated by military applications, slightly before the French Revolution. He raised the following question: what is the cheapest possible way to transport a given pile of sand into a given hole? This seemingly naive optimisation problem turned out to be quite challenging from the mathematical viewpoint, and its resolution had to wait until 1942, when Leonid Kantorovich proposed a solution to a relaxed version of the original problem. Kantorovich received the Nobel prize in Economics, partially because of this work.

This problem had its true renaissance in the late 1980s by the works of Yann Brenier, who realized that this problem in underneath many physical phenomena arising in fluid mechanics, meteorology and elsewhere. In the past 30 years, this theory gained a lot of attention and it became an important branch of pure and applied mathematics. Among others, two recent Fields medalists, Cédric Villani in 2010 and Alessio Figalli in 2018 were recognized for their works in optimal transport.

Beside its deep applications in pure mathematics (ranging from PDEs, probability theory, mathematical physics to metric geometry), this theory turned out to be a very powerful tool to investigive important real life problems arising in machine learning, data science, meteorology, or economics.

The first objective of this project is to understand the mathematical foundations of the optimal transport theory, starting with the one dimensional case. As the second objective, we will consider a variety of recent research papers that propose the study of some interesting economical models through optimal transport. Every participant will write a synthesis on 1-2 such papers, of their choice. Possible subtopics include:

- study of Cournot-Nash equilibria in game theory;

- monopoly pricing model and principal agent problems;

- equilibria in quality markets;

- urban planning and traffic congestion;

- matching problems under transferable utilities; matching problems arising in the the education and labor sectors;

Co/Prerequisites

Main texts

- F. Santambrogio, Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhauser/Springer, Cham, 2015. (A preliminary version of the manuscript is available on the author's webpage here.)

- L. Ambrosio, N. Gigli, A user's guide to optimal transport, lecture notes, online accessible here.

A selection of research papers

- Carlier, G.; Ekeland, I. Equilibrium in quality markets, beyond the transferable case. Econom. Theory 67 (2019), no. 2, 379–391

- Carlier, Guillaume; Zhang, Kelvin Shuangjian, Existence of solutions to principal-agent problems with adverse selection under minimal assumptions. J. Math. Econom. 88 (2020), 64–71.

- Blanchet, Adrien; Carlier, Guillaume, From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2028, 20130398, 11 pp.

- Carlier, Guillaume; Santambrogio, Filippo, A variational model for urban planning with traffic congestion. ESAIM Control Optim. Calc. Var. 11 (2005), no. 4, 595–613.

- Carlier, G.; Jimenez, C.; Santambrogio, F. Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J. Control Optim. 47 (2008), no. 3, 1330–1350

- Blanchet, Adrien; Mossay, Pascal; Santambrogio, Filippo, Existence and uniqueness of equilibrium for a spatial model of social interactions. Internat. Econom. Rev. 57 (2016), no. 1, 31–59.

- Pass, Brendan, Interpolating between matching and hedonic pricing models. Econom. Theory 67 (2019), no. 2, 393–419.

- Erlinger, Alice; McCann, Robert J.; Shi, Xianwen; Siow, Aloysius; Wolthoff, Ronald, Academic wages and pyramid schemes: a mathematical model. J. Funct. Anal. 269 (2015), no. 9, 2709–2746.