# Mini-courses

## 2024 Mini-course I

### Time:

 Lecture 1 Feb 21, Wednesday 10:00-12:00 【Recording】 Lecture 2 Feb 26, Monday 14:00-16:00 【Recording】 Lecture 3 Feb 28, Wednesday 14:00-16:00 【Recording】 Lecture 4 Mar 1, Friday 10:00-12:00 【Recording】

### Syllabus:

• Definitions: Finite metric spaces and graph metrics, l_p metrics, doubling metric spaces, trees, Fréchet embeddings
• Kuratowski Embedding
• Bourgain Embedding Theorem
• Cut-Cone decomposition and isometric embedding of l_2 into L^1
• Goemans-Linial semidefinite relaxation for the Sparsest Cut Problem
• Non-embedding theorems, including Laakso diamond graph
• Embedding simple graphs, e.g. trees. – depending on time.

### Abstract/Description:

This will be a course on metric embeddings to Euclidean space. Algorithms manipulate data, which often can come to us in a complicated form. To enhance our computational abilities, we represent data in different spaces, where computations are easier, via metric embeddings. We will discuss what metric embeddings are and introduce the most important terminology and methods. Then, we see some important classical theorems on existence of embeddings (Assouad, Bourgain and Kuratowski). Given these embeddings, we will discuss their importance to application and tell about the sparsest cut problem and its semi-definite relaxation. Finally, we discuss some ways embeddings may fail to exist, and how simple geometry can help to find embeddings. The focus is on the math, but wherever possible, we make remarks on and give references to more algorithmic results. We may also mention some open mathematical problems of various difficulty.

At the end of the course, by active participation, you should expect to know what metric embeddings are. You would have starting points for further reading and a general view of the field and be able to state why its exciting. You could construct simple embeddings and identify (some) obstructions to embeddability.

Prerequisites: We will be mostly self-contained, but familiarity with the following will be of great help since we will just state these without proof: basic probability (e.g. independence, Chebychev inequality, sums of i.i.d. Normal random variables), basic linear algebra, calculus (Hölder's inequality), familiarity with inner products and norms, graphs and metric spaces (definition enough), basic optimization (e.g. simplex method). This is a mathematical course, so a general familiarity with proofs and a Bachelor's level maturity with mathematics should suffice, which usually includes the previous topics.

## 2023 Mini-course II

### Location and time： L4E48 and zoom

Lecture 1 | August 29, 2023  10:00 - 11:00

Lecture 2 | August 30, 2023  10:00 - 11:00

Lecture 3 | August 31, 2023  10:00 - 11:00

Lecture 4 | September 1, 2023  10:00 - 11:00

### Abstract:

The configuration space Y(X) over a base space X is the space of all Radon point measures on X. The space Y(X) has been studied in many fields such as algebraic geometry (e.g., the hyperplane arrangement with X=Grassmannian), algebraic topology (e.g., the braid group with X=Euclidean plane), representation theory (e.g., the L^2-representation of diffeomorphism groups on manifolds X), statistical physics (e.g., interacting particle diffusions with X=Euclidean space). In this series of lectures, I will focus on the metric geometry of Y(X) induced by the 2-Wasserstein distance. As Y(X) does not support the volume doubling property, the established theory of PI spaces does not apply. The goal of the series is to elaborate on

• Metric geometry on Y(X);
• Curvature analysis on Y(X);
• Applications to infinite particle diffusion processes (including e.g. infinite particle Dyson Brownian motion);
• Open questions.

*After registering, you will receive a confirmation email containing information about joining the meeting.

## 2023 Mini-course I

### Lecture 1   Tuesday, June 20    10 am  【Recording】

Title:  Riemann and Lebesgue Integration
Abstract:  The Riemann integral is perfectly suited for consideration of volume, surface area, arc length, and integration of functions in classical analysis - when the sets in question are smooth and the functions in question continuous.  In this talk, we introduce these ideas and explain the progression from Riemann integration to Lebesgue integration, emphasizing in particular the powerful tools one obtains from this construction.

### Lecture 2  Wednesday, June 21st 10 am　【Recording】

Title:  Capacitary Integration
Abstract:  The Lebesgue integral provides one with a satisfactory tool for many purposes in mathematical analysis.  Yet in the modeling of natural phenomena, with the introduction of partial differential equations, integrals which are not Lebesgue integral makes a prominent appearance - capacitary integrals.  In this talk we discuss this motivation for capacitary integration, with examples, explain the differences with Lebesgue integration, and show the usefulness of these non-standard objects.

### Lecture 3   Thursday, June 22nd 10 am　【Recording】

Title: Capacitary Sobolev Inequalities and Applications
Abstract:  The study of capacities and Capacitary Sobolev Inequalities is now more than half a century old, and yet there are still a number of open research questions to investigate concerning them.  In this talk we discuss in more detail Capacitary Sobolev inequalities with an emphasis on a subject with the most recent activity - Capacitary Sobolev Inequalities around L1.  Open problems will be mentioned.

## 2022 Mini-course I

### 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

### 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.

March 10, 2022  11:00 - 12:00

March 11, 2022  10:00 - 11:00

March 14, 2022  10:00 - 11:00

## 2022 Mini-course III

### 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.

August 22, 2022  15:00 - 17:00

August 23, 2022  15:00 - 17:00

August 24, 2022  15:00 - 17:00