Optimal transport is a well-established field, with applications across the social, life, and physical sciences. Its inventor, Leonod Kantorovich, received the Nobel Prize in Economics (1975). This branch of mathematical theory remains an active area of research, as evidenced by the Fields Medals awarded to Cédric Villani (2010) and Alessio Figalli (2018).

Optimal transport has become an essential tool in various fields where data can be represented as probability distributions, such as images, point clouds, curves, and words. This versatility allows it to power a range of techniques, from clustering to dimensionality reduction. In recent years, it has become a pivotal instrument in the context of machine learning research. For example, when analyzing a group of related data distributions controlled by certain parameters, geometrical properties of optimal transport enable the model to capture both local and global relationships within the data, making it capable of seeing intricate patterns and dependencies. A notable application is the Wasserstein barycenter, which provides an average of data samples which respects the geometric properties of the data and facilitates impressive image transformations. The Wasserstein barycenter between multiple images produces a visually appealing and semantically meaningful result.

This Summer School aims to give an introduction to the research area of optimal transport by exploring the analytical, probabilistic, geometrical, computational, and statistical aspects of the theory. Applications in natural sciences (e.g., theoretical chemistry, computational biology) and machine learning may also be discussed.

Prerequisites: advanced-level courses in probability, mathematical analysis, and linear algebra. Basic knowledge of functional analysis and familiarity with Python would be beneficial, but not mandatory. Out of the five courses listed below, two will be challenging but specifically tailored to undergraduate students.