Math as an Adventure: The Unexpected Applications of the Optimal Transport Problem

Think of a company that needs to transport some goods from its production plants, each of which has produced a given fraction of the total, to its warehouses, each of which must receive a desired quantity. Given the transportation costs between each plant and each warehouse, which is the transportation plan corresponding to the smallest total cost?

This is a simple version of the optimal transport problem, which can be formulated also in more abstract and complex settings. It is a classic mathematical problem, first formalized by the French mathematician Gaspard Monge in 1781. Major advances were made by the Russian mathematician Leonid Kantorovich during World War II, but were published only in the 1960s, due to the limitations imposed by the USSR on such strategic research topics. In fact, Kantorovich’s work on optimal transport was so relevant that in 1975 it earned him the Nobel prize for economy.

From its economic roots, optimal transport then grew far more general, with applications in various fields, ranging from geometry to probability, statistics, partial differential equations and – more recently – machine learning. «This is a nice example of the power of a mathematical approach», explains Giuseppe Savaré, Professor of Mathematical Analysis at Bocconi Department of Decision Sciences. «Abstracting from the initial applied problem and deriving a more general formulation often opens up unexpected research directions, contributing to apparently unrelated topics. From time to time, I find out that some results that we obtained years ago are now being employed in machine learning or image processing, for applications that did not even exist back then, at least not in their current form».

«Twenty years ago, when I started working on optimal transport and gradient flows with Luigi Ambrosio and Nicola Gigli, we actually did something similar. We combined some mathematical ideas on gradient flows – that had been formulated in the 1980s in a different and quite abstract context – with the newest developments of optimal transport. Such a dynamical approach was the starting point of our investigation. This again proves the importance of basic research: even when applications are not immediately visible, general and elegant ideas can prove fundamental in the longer term. Moreover, it shows a beautiful and exciting side of mathematics: when you start working on a new challenging problem, you never know exactly where you will end up».

Find out more

L. Ambrosio, N. Gigli and G. Savaré. Gradient flows: in metric spaces and in the space of probability measures. Birkhäuser, 2008.

Videos (in italian):

G. Savaré. Conferenza su "Trasporto ottimo, flussi gradienti e geometria Riemanniana", XXI Congresso dell’Unione Matematica Italiana. Pavia 2-7 settembre 2019.

Intervista a G. Savaré per il XXI Congresso dell’Unione Matematica Italiana. Pavia 2-7 settembre 2019.

by Sirio Legramanti

Source of the article: Bocconi Knowledge