From Combinatorics to Partial Differential Equations

TL;DR

This paper explores the connection between combinatorial optimal matching problems in point clouds and partial differential equations, emphasizing the critical role of dimension and applications in statistics.

Contribution

It introduces an analytical perspective linking optimal transportation to PDEs in the context of random point clouds, highlighting the dimension-dependent behavior.

Findings

Dimension $d=2$ is critical for the problem.

Analytical methods connect combinatorics to PDEs.

Applications in statistical analysis of point clouds.

Abstract

The optimal matching of point clouds in $\mathbb{R}^d$ is a combinatorial problem; applications in statistics motivate to consider random point clouds, like the Poisson point process. There is a crucial dependance on dimension $d$, with $d=2$ being the critical dimension. This is revealed by adopting an analytical perspective, connecting e.\,g.~to Optimal Transportation. These short notes provide an introduction to the subject. The material presented here is based on a series of lectures held at the International Max Planck Research School during the summer semester 2022. Recordings of the lectures are available at https://www.mis.mpg.de/events/event/imprs-ringvorlesung-summer-semester-2022.