N.B. These sessions will take place online via zoom: https://ubc.zoom.us/j/69961568613?pwd=eUJ4SFRneGIxbmdGUkQ5OEFLeHhudz09
Inspired by Perelman’s work on the entropy formula for Ricci flow, we introduce the $W$-entropy and prove its monotonicity and rigidity theorem for the geodesics flow on the Wasserstein space over Riemannian manifold. This improves an earlier result due to Lott and Villani on the displacement of the Boltzmann type entropy on Riemannian manifolds with non-negative Ricci curvature. We then introduce the Langevin deformation on the Wasserstein space, which interpolates the Wasserstein geodesic flow and the gradient flow of the Boltzmann entropy (i.e., the heat equation on the underlying manifold). Moreover, we present a $W$-entropy type formula for the Langevin deformation. Joint work with Songzi Li.
Optimal transport studies the most economical movement of resources. In other words, one considers a pile of raw material and wants to transport it to a final configuration in a cost-efficient way. Under quite general assumptions, the solution to this problem will be induced by a transport map where the mass at each point in the initial distribution is sent to a unique point in the target distribution. In this talk, we will discuss the regularity of this transport map (i.e., whether nearby points in the first pile are sent to nearby points in the second pile). $$ $$ It turns out there are both local and global obstructions to establishing smoothness for the transport. When the cost is induced by a convex potential, we show that the local obstruction corresponds to the curvature of an associated Kähler manifold and discuss the geometry of this curvature tensor. In particular, we show (somewhat surprisingly) that its negativity is preserved along Kähler-Ricci flow.
We discuss the gradient flow structure of hydrodynamic limit equations obtained from microscopic interacting particle systems. We first show that for a wide class of reversible non-degenerate interacting particle systems, their hydrodynamic limit equations are formally given as a gradient flow with respect to the Wasserstein metric with mobility. Then we discuss when such formulation can be justified rigorously, and how it can be applied for the study of hydrodynamic limits. This talk is based on an ongoing work with Kohei Hayashi.
Multi-marginal optimal transport, a natural extension of the well-known classical optimal transport problem, is the problem of correlating given probability measures as efficiently as possible relative to a given cost function. Although a variety of applications have arisen over the past twelve years, the structure of solutions for the multi-marginal case has been difficult to address, mainly due to the strong dependence on the cost function. In this talk, I will briefly outline the known theory for uniqueness of this problem. Next, I will present a recent joint work with Brendan Pass based on a general condition on the cost function that provides uniqueness.
Consider the problem of matching two independent sets of $N$ i.i.d. observations from two densities $\rho_0$ and $\rho_1$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with $N$ atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as $N\rightarrow \infty$, to the solution of a variational problem, introduced by Föllmer, called the Schrödinger problem. Finally, we prove limiting Gaussian fluctuations for this convergence in the form of Central Limit Theorems for integrated test functions. This is enabled by a novel chaos decomposition for permutation symmetric statistics, generalizing the Hoeffding decomposition for U-statistics. Our results establish a novel passage for the transition from discrete to continuum in Schrödinger’s lazy gas experiment.
In this talk, we will introduce some interesting applications of optimal transportation in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric inequality in geometry; and an application in image recognition relating to a transport between hypercubes. This talk is based on a series of joint work with Shibing Chen, Xu-Jia Wang, and with Gregoire Loeper.