## Student Probability Seminar, Fall 2019

### Wednesday 2-3 PM, 939 Evans

This is the UC Berkeley Student Probability Seminar, run by students in both the Mathematics and Statistics Departments at UC Berkeley. We meet on Wednesdays in 939 Evans from 2-3pm, right before the probability seminar.

This semester we will be reading Optimal Transport: Old and New by Cédric Villani. The book is available for free (on campus through Springer Link) here, and there is also a free copy available everywhere here. We may also have talks on other probability topics by students, but this will be at a different time and will be announced as they happen.

Previous Seminars: Spring 2019 , Fall 2018 , Summer 2017 , Fall 2017
 DATE SPEAKER TITLE (click for abstract below) August 28 Milind Hegde Logistics of the seminar & Intro material (Ch 1 & 4) September 4 Efe Aras Cyclic Monotonicity, c-Convexity, and the Duality principle September 11 Zsolt Bartha The Kantorovich duality theorem and some special cases September 18 Ella Hiesmayr Equivalences of optimality of transferance plans September 25 Efe Aras Wasserstein distances October 2 Adam Jaffe Wasserstein distances (cont'd) October 9 N/A Cancelled October 16 Zhiyi You Displacement interpolation October 23 Ian Francis Displacement interpolation (cont'd) October 30 Alexander Tsigler Displacement interpolation (cont'd) November 6 Izak Oltman Displacement interpolation (cont'd) November 13 Milind Hegde Displacement interpolation (cont'd) November 20 Efe Aras Some properties of optimal couplings November 27 N/A Thanksgiving break December 4 Farzad Pourbabaee Optimal transport applications in economics

### Logistics of seminar & Intro material

Milind Hegde

We will talk about what a coupling of two probability measures is, give some examples from probability theory, and state some simple tools that will be useful throughout the semester (chapter 1). Then we will introduce the basic problem of optimal transport, and prove that there exists an optimal coupling (chapter 4).

### Cyclic Monotonicity, c-Convexity, and the Duality principle

Efe Aras

Today, for the first half, we will set up notions of cyclical monotonicity, and a generalized notion of convexity (called c-convexity). We will then start on the big theorem of Chapter 5, which is a central duality principle in optimal transport. We will start proving it by relying on properties of cyclically monotone sets.

### The Kantorovich duality theorem and some special cases

Zsolt Bartha

Today we will continue with the proof of the Kantorovich duality theorem, and will point out some interesting special cases.

### Equivalences of optimality of transferance plans

Ella Hiesmayr

We will finish the proof of Theorem 5.10, in particular we will show several equivalences of the optimality of a transference plans for real-valued cost functions.

### Wasserstein distances (cont'd)

We will focus on basic properties of Wasserstein distances and Wasserstein spaces, with an eye towards topology. In particular, we will characterize the relationship between convergence in the Wasserstein metric and weak convergence, and we will prove that $$P_p(X)$$ is complete and separable whenever $$X$$ is complete and separable.

### Displacement interpolation

Zhiyi You

We will now discuss a time-dependent version of optimal transport leading to a continuous displacement of measures, which gives a more complete description of the transport and a richer mathematical structure for future use. Several new concepts, including action and displacement interpolation, will be introduced along with examples and informal discussions before we move forward.

### Displacement interpolation (cont'd)

Ian Francis

We will continue discussing displacement interpolation. Specifically, we will quickly review deterministic interpolation, prove some basic properties of Lagrangian actions, and finally introduce interpolation of random variables.

### Displacement interpolation (cont'd)

Alexander Tsigler

We will revise the properties of Lagrangian actions from the last time, and prove some of them. If time permits, we'll state the "main theorem" of displacement interpolation (Theorem 7.21) and discuss an outline of its proof.

### Displacement interpolation (cont'd)

Izak Oltman

We will discuss the main theorem on displacement interpolation (7.21), give an outline of the proof, and prove two corollaries. If time permits, we will discuss displacement interpolation between intermediate times and restriction.

### Displacement interpolation (cont'd)

Milind Hegde

We will discuss the main theorem on displacement interpolation (Theorem 7.21) and give its proof.

### Some properties of optimal couplings

Efe Aras

We will talk about some useful properties of optimal couplings; for instance, there is an optimal map between two sufficiently smooth densities. We will also discuss an example where the optimal transport map cannot be smooth. We will also note that between two log-concave distributions, the optimal transport map is Lipschitz, and, if we have time, we will explain how that fact allows transporting inequalities from one random variable to another.