Lecturer

Dr. Mikhail Chebunin

Teaching Assistant

Dr. Fausto Colantoni

Time and Place

Lecture

TBA

Exercise

TBA

Format and Language

2+1 (2 hours lecture + 1 hour exercise session per week), 42 hours (28 lecture hours + 14 exercise hours). The course will be taught in English.

Requirements

Students should be familiar with basic courses in Analysis and Probability. Also helpful, but not required, is basic knowledge in Functional and Convex analysis.

Target groups

Master Math, DaSci, MaBi, Fin, WiMa, CSE.

Description

Optimal transport is a mathematical theory that connects probability, optimization, functional analysis, and convex analysis. Originally, it was introduced by Monge in 1781 as a problem of optimally transporting soil, and then became a fundamental tool in modern probability, machine learning, and data science. This course provides an introduction to modern foundations/background in optimal transport theory with applications to statistics, machine learning, and data science.

We begin with finite-dimensional problems to build intuition, then develop the general theory on Polish spaces using tools from probability theory (couplings, pushforward measures, weak convergence). Core topics include the Kantorovich relaxation and duality, Brenier's theorem, and Wasserstein distances. The course emphasizes connections to analysis and probability, including couplings as transport plans, Wasserstein metrics as natural distances on probability spaces, and applications to empirical measures and statistical estimation.

The course mostly follows the recent book by Gero Friesecke (SIAM, 2024), which provides a modern, accessible treatment. Students will develop both theoretical and practical skills, including R/Python programming focus on data science problems.

Topics:

• Optimal Transport on Finite Spaces

• Measures on Polish Spaces

• Existence of Optimal Transport Plans

• Kantorovich Duality

• Brenier's Theorem and Optimal Maps

• Wasserstein Distances

• Entropic Regularization and Computation

• Selected Applications, e.g., statistical applications, Wasserstein GANs in machine learning, and applications to data science. 
   

Exam

Final written or oral examination, depending on the number of participants. The examination form will be announced in advance, at least 4 weeks before the examination date. The prerequisite for taking the exam is to achieve at least 50 % of the practice points.

Exercise sessions

Problem-solving sessions emphasizing computational examples and analysis-probabilistic proofs. Exercise sheets will be distributed every one or two weeks in the Moodle course.

Literature

• Friesecke, Gero. Optimal Transport: A Comprehensive Introduction to Modeling, Analysis, Simulation, Applications, SIAM, 2024.

• Santambrogio, Filippo. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Birkhäuser, 2015.

• Peyré, Gabriel and Cuturi, Marco. Computational Optimal Transport, Foundations and Trends in Machine Learning, 2019.