Geometric Methods in Partial Differential Equations

TL;DR

This paper explores the relationship between geometry and partial differential equations, emphasizing the importance of algebraic geometry tools like the Cayley-Bacharach theorem in analyzing solution spaces.

Contribution

It introduces a geometric framework for PDE analysis that leverages classical algebraic geometry results to understand solution space dimensions.

Findings

Connections between algebraic geometry and PDE solution spaces

Application of Cayley-Bacharach theorem to PDE analysis

Enhanced understanding of zero sets of differential operators

Abstract

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret's theorem, and the Brill-Noether Restsatz theorem.