Preparation minischool for CMC conference: "Analysis, Geometry, and Optimal Transport”

Speaker: Seung-Yeal Ha
Title: Collective dynamics of interacting many-body systems
Abstract: Collective phenomena such as flocking and synchronization are ubiquitous in our natural systems, e.g., the aggregation of bacteria, flocking of birds, swarming of fish, herding of sheep, etc. For these collective dynamics, many phenomenological models were proposed and studied in many scientific areas such as control theory, mathematical biology, statistical physics. In my three lectures, I will survey a recent mathematical progress for flocking and synchronization. The three hour lecture plans are as follows:

I) Overview
II) Emergent dynamics of the Cucker-Smale model for flocking
III) Emergent dynamics of the Kuramoto model for synchronization

Speaker: Young-Heon Kim
Title: Some basics of optimal transport
Abstract: Optimal transport is about matching mass distributions in an efficient way. It is a rapidly developing subject, with applications and connections to various areas in pure and applied mathematics, including geometric and functional inequalities, convex geometry, geometry of Ricci curvature, analysis of PDE’s, probability,  mathematical finance, to name a few. The aim of these lectures is to explain some basic concepts of optimal transport to nonspecialists. The following topics will be covered:

I) Optimal transport with the quadratic cost: Duality and Monge Ampere equation
II) W_2 metric, displacement interpolation and displacement convexity
III) Regularity of Monge Ampere equation: some basics

Speaker: Filippo Santambrogio
Title: Optimal transport methods for gradient-flow evolution PDEs
Abstract: In this 4h mini-course, i will first review the notion of gradient flow in the easiest case, i.e. $x'(t)=-nabla F(x(t))$ in $mathbb{R}^n$, and its variational approximation, by the minimizing movement scheme where one iteratively solves $$x_{k+1}in argmin F(x)+frac{|x-x_k|^2}{2tau}$$ for a fixed time-step $tau>0$.
Then I will show how this approximation (which actually amounts to an implicit Euler scheme) can be used for evolution in metric spaces and show what it gives when one uses the metric space of probability measures (on a given domain that will be supposed to be compact for simplicity) endowed with the Wasserstein distance $W_2$ arising from optimal transport (and I will strongly use some of the notions introduced in the course by Y.-H. Kim). This uses the so-called Jordan-Kinderlehrer-Otto scheme: an idea of the methods to prove the convergence of this scheme will be quickly evoked.
After identifying the equations that can be studied in this way (which include the heat equation, Fokker-Planck, porous media and other interesting cases) we will use the JKO scheme to prove estimates on the solution. In particular we will consider $L^infty$ estimates, which require the use of the Monge-Amp`ere equation, and some $BV$ or $W^{1,p}$ estimates, which require a new inequality on optimal transport.