Optimal Transport and Congestion
Short course in the MFO seminar on Optimal Transport Theory and Hydrodynamics
This course, part of the Oberwolfach Seminar held from October 14 to
19 at MFO, contains four lectures.
1) (15/10, morning) Introduction to static OT.
Monge and Kantorovich problems, dual problem, optimality conditions,
Brenier theorem, 1D case.
2) (15/10, afternoon) Introduction to dynamic OT.
Wasserstein distances, curves in the Wasserstein space, geodesics,
3) (16/10, morning) Stationary traffic congestion.
Beckmann problem, transport density, measures on curves. Nash and Wardrop
equilibria and traffic congestion.
4) (16/10, morning) Mean Field Games.
Introduction to MFG and in particular variational
MFG. Duality. Optimality conditions and Lagrangian equilibria. Few
words on regularity.
For the two first lectures the main reference is my book
Optimal Transport for Applied Mathematicians (see here or here for a non-official version).
Lecture 1 : Chapter 1, sections 1.1 to 1.3
Lecture 2 : Chapter 5
Lecture 3 uses Chapter 4 from this book, but one can also look at this survey
The topics of
Lecture 4 are only mentioned in the book (section 8.4.4), but one can also look at this survey