Universität Wien

250119 VO Optimal Transport (2025W)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 01.10. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 06.10. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.10. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 13.10. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.10. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 20.10. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 22.10. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 27.10. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.10. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 03.11. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 05.11. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 10.11. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 12.11. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 17.11. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.11. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 24.11. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.11. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 01.12. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 03.12. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 10.12. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 15.12. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 17.12. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 07.01. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 12.01. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 14.01. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 19.01. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 21.01. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 26.01. 08:00 - 09:30 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 28.01. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß

Information

Aims, contents and method of the course

The theory of Optimal Transport finds its roots in the work of Gaspard Monge in 1781, whose interest was in finding the cheapest possible way to transfer some resources from producers to consumers. After some major contributions due to Leonid Kantorovich (who was later awarded the Nobel Prize in Economics, partly for his work on the subject) in the 1940s, Optimal Transport has been blooming in the last thirty years, with applications covering partial differential equations, geometric analysis, mathematical finance, and machine learning, among others.

The goal of this course is to introduce the basic theory of Optimal Transport and to hint at some of its more recent developments and applications.
The main topics covered will include:
- classical formulations of the problem;
- duality theory;
- optimality conditions;
- existence of optimal transport maps (Brenier’s theorem);
- the Monge-Ampère equation;
- applications to isoperimetric inequalities;
- the Wasserstein distance and the Wasserstein space;
-dynamical formulation of optimal transport, continuity equation and Benamou-Brenier formula;
-displacement convexity and applications.

Confidence with abstract measure theory and integration (as covered for instance in Chapter 1 of Evans and Gariepy's book "Measure Theory and fine properties of functions" or Chapters 1-3 and 7-8 of Rudin's book "Real and Complex Analysis"), basic functional analysis (duality, weak convergence, Banach-Alaoglu's theorem), regularization by convolution, and classical tools of analysis in several variables (change of variables formula, Gauss-Green and divergence theorems) is expected, although many preliminary results will be stated and briefly discussed in class .

Assessment and permitted materials

Oral exam (30 minutes), to be scheduled flexibly.

Minimum requirements and assessment criteria

Examination topics

All content discussed in class is examinable, unless explicitly declared otherwise.

Reading list

Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe: Gradient flows in metric spaces and in the space of probability measures.
2nd ed. Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser vii, 334 p. (2008).

Figalli, Alessio; Glaudo, Federico: An invitation to optimal transport, Wasserstein distances, and gradient flows.
EMS Textbooks in Mathematics. Berlin: European Mathematical Society (EMS) vi, 136 p. (2021).

Villani, Cédric: Topics in optimal transportation.
Graduate Studies in Mathematics 58. Providence, RI: American Mathematical Society (AMS) xvi, 370 p. (2003).

Ambrosio, Luigi; Bruè, Elia; Semola, Daniele: Lectures on optimal transport.
Unitext, 169, La Mat. per il 3+2, Springer, Cham, (2024), xi+260 pp.

Association in the course directory

MGEV; MANV; MSTV; ML2; MEL

Last modified: We 20.05.2026 15:47