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, Benamou-Brenier formaula.
  • 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). More precisely:
    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