Contact manifolds, Lagrangian Grassmannians and PDEs

TL;DR

This paper reviews a geometric framework for analyzing scalar PDEs using contact manifolds and Lagrangian Grassmannians, providing an accessible introduction with references to advanced research topics.

Contribution

It offers a comprehensive overview of the geometric approach to PDEs, emphasizing the role of contact manifolds and Lagrangian Grassmannians, suitable for graduate students and researchers.

Findings

Introduces geometric methods for scalar PDEs

Connects contact geometry with PDE theory

Provides references to current research topics

Abstract

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a 30-hours Ph.D course given by two of the authors (GM and GM). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.