401-3350-68L  Introduction to Optimal Transport

SemesterHerbstsemester 2018
DozierendeA. Figalli, weitere Referent/innen
Periodizitäteinmalige Veranstaltung
LehrspracheEnglisch
KommentarMaximale Teilnehmerzahl: 11



Lehrveranstaltungen

NummerTitelUmfangDozierende
401-3350-68 SIntroduction to Optimal Transport
Advisors: F. Glaudo and G. Franz
2 Std.
Mo10:15-12:00ETZ E 6 »
A. Figalli, weitere Referent/innen

Katalogdaten

KurzbeschreibungIntroductory seminar about the theory of optimal transport.
Starting from Monge's and Kantorovich's statements of the optimal transport problem, we will investigate the theory of duality necessary to prove the fundamental Brenier's theorem.
After some applications, we will study the properties of the Wasserstein space and we will conclude introducing the dynamical point of view on the problem.
Lernziel
InhaltGiven two distributions of mass, it is natural to ask ourselves what is the "best way" to transport one into the other. What are mathematically acceptable definitions of "distributions of mass" and "to transport one into the other"?
Measures are perfectly suited to play the role of the distributions of mass, whereas a map that pushes-forward one measure into the other is the equivalent of transporting the distributions. By "best way" we mean that we want to minimize the map in some norm.

The original problem of Monge is to understand whether there is an optimal map and to study its properties. In order to attack the problem we will need to relax the formulation (Kantorovich's statement) and to apply a little bit of duality theory. The main theorem we will prove in this direction is Brenier's theorem that answers positively to the existence problem of optimal maps (under certain conditions).
The Helmotz's decomposition and the isoperimetric inequality will then follow rather easily as applications of the theory.
Finally, we will see how the optimal transport problem gives a natural way to define a distance on the space of probabilities (Wasserstein distance) and we will study some of its properties.
Literatur"Optimal Transport, Old and New", C. Villani
[Link]

"Optimal Transport for Applied Mathematicians", F. Santambrogio
[Link]
Voraussetzungen / BesonderesThe students are expected to have mastered the content of the first two
years taught at ETH, especially Measure Theory.
The seminar is mainly intended for Bachelor students.

In order to obtain the 4 credit points, each student is expected to give two 1h-talks and regularly attend the seminar. Moreover some exercises will be assigned.

Further information can be found at Link

Leistungskontrolle

Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Leistungskontrolle als Semesterkurs
ECTS Kreditpunkte4 KP
PrüfendeA. Figalli
Formunbenotete Semesterleistung
PrüfungsspracheEnglisch
RepetitionRepetition nur nach erneuter Belegung der Lerneinheit möglich.

Lernmaterialien

 
HauptlinkInformation
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Einschränkungen

PlätzeMaximal 11
VorrangDie Belegung der Lerneinheit ist bis 30.12.2018 nur durch die primäre Zielgruppe möglich
Primäre ZielgruppeMathematik BSc (404000) ab Semester 05
Mathematik MSc (437000)
Angewandte Mathematik MSc (437100)
WartelisteBis 31.12.2018

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StudiengangBereichTyp
Mathematik BachelorSeminareWInformation
Mathematik MasterSeminareWInformation