# Mini-courses

## 2022 Mini-course I

### Title: Curvature and Optimal transport

### Speaker: Professor Asuka Takatsu, Tokyo Metropolitan University

In this series of lectures, I first review the notion of curvature (Gaussian curvature and Ricci curvature).

In particular, I recall some comparison theorems (Toponogov's triangle comparison theorem, Bishop--Gromov volume comparison theorem etc).Then I introduce a generalized notion of curvature in non-smooth spaces.

March 8, 2022 10:00 - 11:00 AM

March 9, 2022 10:00 - 11:00 AM

March 10, 2022 9:30 - 10:30 AM

## 2022 Mini-course II

### Title: A brief introduction to branched optimal transport

### Speaker: Professor Jun Kitagawa, Michigan State University

The optimal transport (also known as Monge-Kantorovich) problem is a classical optimization problem which has recently become the focus of much research with connections to various fields such as PDEs, geometry, and applications. In particular, it provides an effective way to metrize the space of probability measures on a given metric space. However, there is an alternate approach to metrizing such spaces using so called branched optimal transport. Branched optimal transport is based on the classical Gilbert-Steiner problem, later adapted by Qinglan Xia, and in contrast to the Monge-Kantorovich approach tends to yield branching structures. In this series of lectures I will introduce the basics of branched optimal transport and discuss some of the known results in the literature.

## 2022 Mini-course III

### Title: BV functions in Carnot groups

### Speaker: **Dr. Sebastiano** Nicolussi Golo, University of Jyväskylä

In this lectures I will present the theory of functions of bounded variation (BV functions) in Carnot groups. Carnot groups are Lie groups endowed with a fractal sub-Riemannian metric structure. The exposition will always reference to the classical theory of BV functions on Euclidean spaces (see for instance the monograph by Ambrosio-Fusco-Pallara). In this way, I expect that my lessons will be informative also for those who are familiar neither with BV functions on Euclidean spaces nor with Carnot groups.

In particular, we will study sets of finite perimeter. The structure of sets of finite perimeter is a cornerstone of Geometric Measure Theory. We will see how it (does not yet) extend to Carnot groups.