Topics


Here is the (aspirational) list of topics to be covered this semester, individual items are not necessarily of comparable length in terms of number of lectures . For instance, item #2 is probably the single longest item requiring 2-3 weeks, while item #5 should take at most a week. A majority of items below will take one or two lectures.

  1. The power of optimal transport via three examples
  2. The Monge-Kantorovich problem (setup, existence, duality, $c$-cyclical monotonicity, $c$-convexity)
  3. Brenier's theorem and the Monge-Ampere equation
  4. A quick overview of the regularity theory for the Monge-Ampere equation
  5. Minimizing movements and gradient descent
  6. Caffarelli's theorem for log-concave densities
  7. The transport (Wasserstein) metric and gradient flows in $P_2$
  8. The Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation
  9. Statistical estimation of transport distances
  10. Gromov-Wasserstein distances (regularization, duality, sampling)
  11. Gradient flows and transformers