Print syllabusTransportation of measure
General data
| Course ID: | 1000-1M20TM |
| Erasmus code / ISCED: |
11.1
|
| Course title: | Transportation of measure |
| Name in Polish: | Transport miary |
| Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
| Course groups: |
(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics |
| ECTS credit allocation (and other scores): |
(not available)
|
| Language: | English |
| Type of course: | elective monographs |
| Requirements: | (in Polish) Wstęp do procesów stochastycznych 1000-135WPS |
| Prerequisites: | (in Polish) Wstęp do procesów stochastycznych 1000-135WPS |
| Prerequisites (description): | The requirements are: good knowledge of Measure Theoy and Probability Theory as well as basic knowledge of Functional Analysis, Partial Differential Equations and Stochastic Processes. |
| Short description: |
The course will be devoted to basic results of the theory of transportation of measure with focus on its probabilistic aspects, in particular on connections with the theory of concentration of measure. |
| Full description: |
The theory of optimal transportation, initiated at the end of the 18th century by Monge, concerns the ways of moving resources in space in a way, which minimizes the total cost for a given initial and final distribution of the resources. Mathematically the problem is usually modelled in terms of probability measures and the task is to find a map which transports one measure onto another minimizing a certain cost, defined in terms of an appropriate integral functional (in one of the simplest formulations the cost may be the average distance by which an individual point is moved). Over the years the theory of optimal transportation has found many applications, both in theoretical mathematics and in economics. Since the formulation of the basic problem, the theory has undergone several periods of rapid developments, in particular in the 1940s (thanks to the work by Kantorovich - Nobel Prize in Economics 1975) and in the last 30 years (thanks to the work by, e.g., Cedric Villani - Fields Medal 2010, and Alesio Figalli - Fields Medal 2018). It has deep connections with many branches of mathematics: with partial differential equations, probability theory and differential geometry. During the course I will present the foundations of the theory as well as selected connections with the aforementioned fields of mathematics. The lectures will be focused on probabilistic aspects, however many important tools will come from partial differential equations or functional analysis. In particular I will present - Monge's problem - general transportation costs, Kantorovich-Wasserstein distances - Kantorovich duality - Brenier's theorem - the Benamou-Brenier formula and connections with the transport equation - Talagrand's theorem, transportation inequalities, connections with the concentration of measure phenomenon - the Otto-Villani theorem, connections with logarithmic Sobolev inequalities - weak transport costs - the Caffarelli principle - martingale transportation or entropic transportation Most of the topics will be presented with full proofs, while some of more advanced theorems will be accompanied by sketches of the arguments. |
| Bibliography: |
- Cedric Villani, Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003 - Cedric Villani, Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. - Stéphane Boucheron, Gábor Lugosi, Pascal Massart, Concentration inequalities. A nonasymptotic theory of independence. With a foreword by Michel Ledoux. Oxford University Press, Oxford, 2013 - Michel Ledoux, The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. - Dominique Bakry, Ivan Gentil, Michel Ledoux, Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften, 348. Springer, Cham, 2014. |
| Learning outcomes: |
Students: - know the mathematical formulation of the problem of optimal transportation and understand the underlying intuition coming from applied sciences - are able to define various costs appearing in applications - are able to apply the Kantorovich theorem to state the dual formulation of the transport problem - are able to prove Brenier's theorem - know the connections of the theory of optimal transportation with the transport equation and with the Hamilton-Jacobi equations - are familiar with basic concentration inequalities and can prove them using the transportation approach. |
| Assessment methods and assessment criteria: |
The final grade will be based on a take-home exam and in individual cases on a conversation concerning solutions. |
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