Group meets at 2:30pm on Tuesdays in Wean Hall 7218.
- January 26th at 2:30pm: Dejan Slepcev “Intro to Kantorovich description of optimal transportation and convex duality” [ slides similar to what was covered]
- February 2nd at 2:30pm: Ryan Murray “Kantorovich duality and existence of optimal transportation plans and maps” [Chapter one in Villani, Chapter one in Brezis] Lecture notes: OT-Duality
- February 9th at 2:30pm: (Ryan will finish Kantorovich duality first, then) Matt Thorpe “Brenier’s theorem and characterization of OT maps” [Chapter 2 in Villani] Lecture notes: Characterisation of OT Maps
- February 16th at 2:30pm: Matt Thorpe will finish Brenier’s theorem.
- February 23th at 2:30pm: David Kinderlehrer will discuss his paper with Jordan and Otto on gradient flows and the connections with the Fokker-Planck equation (see the papers to read section for the reference).
- March 1st at 2:30pm: David will finish presenting his paper “the variational formulation of the Fokker-Planck equation” where the Wasserstein distance with entropic regualization is used to define a gradient flow for the Fokker-Planck equation.
- March 15th, 2:30pm: Slav Kirov will present on numerical approaches to optimal transportation based on linear programming and regularizations. There have been exciting recent developments in this area. References include papers by Cuturi, Benamou, Carlier, Cuturi, Nenna, and Peyre, Oberman and Ruan, Merigot and Oudet.
- March 22nd, 2:30pm: Marco Caroccia will present on geometry of optimal transportation and in particular on showing that it can formally be seen as a Riemannian manifold. The geodesic distance on the manifold is the Wasserstein distance. This follows from the work of the Benamou and Brenier which will be presented. Furthermore modern understanding as to what is the structure of the manifold (say the tangent space) will be discussed. Notes are here.
- March 29, 2:30pm: Antoine Remond-Tiedrez will discuss connections of optimal transport and action minimization (kinetic energy as in Brenier-Benamou formula) with Euler’s equation. This was by discussed by Arnold for classical solutions and was used by Brenier to introduce generalized solutions. Antoine will mainly focus on discretization of such solutions recently introduced by Merigot and Mirebeau.
- April 5, 2:30pm: Antoine Redmond-Tiedrez will continue with the discussion on the connections of optimal transport and action minimization.
- April 12, 2:30pm: Matt Thorpe will give the background on an approach to explaining the microscopic origins of gradient flows. A large deviation principle explains why one takes the energy minimizing path in a gradient flow. Riccardo Cristoferi will continue next week. Notes can be found here.
- April 19, 2:30pm: Riccardo Cristoferi will continue with the microscopic origins of gradient flows via a large deviation principle. Instead of considering a continuum model, e.g. the Heat equation, one can consider a family of particles that evolve via a diffusion in such a way that the empirical measure approximates the continuum model, e.g. Brownian motion. When the distribution of the particles satisfies a large deviation principle then one can ask how this relates to a gradient flow on the continuum model. The paper by Adams, Dirr, Peletier and Zimmer gives an explicit relationship between the large deviation rate function and the gradient flow formulation. Notes here [will be updated soon].
- April 26, 3:00pm ***note the change in time***: Kevin Ou will talk about transport problems which penalize the path. In the case of a concave cost as function of quantity, Kantorovich’s formulation of optimal transport no longer captures the main features of reality and fails to generate the real world optimal transport path. Rather than considering the transport plan, we directly minimize cost functionals on transport paths, starting from atomic probabilities of sources and targets, formulating it in the language of geometric measure theory, and generalizing the formulation and results to the entire space of probabilities. The main results are the existence of the optimal transport paths (branched optimal transport), and a new metrization of the weak-star topology on probabilities in terms of branched optimal transport cost. Furthermore, this new metric space is geodesic. Emphases are on the formulation of the problem and the intuitions behind these results. The main reference is Optimal paths related to transport problems, Qinglan Xia.
- May 3, 2:30pm: Kevin Ou will continue discussing branched transport. Full notes are here: ou_introduction_branched_transport.