Derived Geometry and Non-Linear Differential Equations on the Punctured Disc

TL;DR

This paper explores the derived geometric structures of non-linear differential equations on the punctured disc, revealing a deep connection between variational principles and symplectic forms in derived solution spaces.

Contribution

It introduces a novel derived geometric framework for analyzing non-linear differential equations, linking variational formulations to residue pairings and symplectic structures.

Findings

Derived spaces of solutions carry a (-1)-symplectic form.

Variational formulations are equivalent to residue pairings.

Provides a new perspective on non-linear differential equations via derived geometry.

Abstract

We study non-linear differential equations on the punctured formal disc by considering the natural derived enhancements of their spaces of solutions. In particular, by appealing to results of the inverse theory in the calculus of variations, we show that a variational formulation of a differential equation is \emph{equivalent} to the residue pairing inducing a (-1)-symplectic form on the derived space of solutions equipped with a certain decoration of its tangent complex.