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Representability in non-linear elliptic Fredholm analysis

John Pardon

TL;DR

This work develops a derived differential-geometric framework for representing moduli spaces of solutions to non-linear elliptic Fredholm PDEs, focusing on pseudo-holomorphic maps. It proves representability of the moduli functor on the ∞-category of derived smooth manifolds $\mathsf{Der}$ via a three-step argument: establish representability on the regular locus using standard Fredholm theory, extend to derived smooth settings through left Kan extension, and recover full representability by a thickening argument. The approach aims to provide a canonical, atlas-free perspective analogous to moduli functors in algebraic geometry and discusses extending the framework to degenerations using log smooth manifolds. Together, these ideas lay groundwork for a virtual fundamental geometrical theory in derived and log-smooth settings, potentially unifying approaches to enumerative problems in symplectic and complex geometry.

Abstract

We summarize current work aimed at showing that moduli spaces of solutions to non-linear elliptic Fredholm partial differential equations are derived log smooth manifolds.

Representability in non-linear elliptic Fredholm analysis

TL;DR

This work develops a derived differential-geometric framework for representing moduli spaces of solutions to non-linear elliptic Fredholm PDEs, focusing on pseudo-holomorphic maps. It proves representability of the moduli functor on the ∞-category of derived smooth manifolds via a three-step argument: establish representability on the regular locus using standard Fredholm theory, extend to derived smooth settings through left Kan extension, and recover full representability by a thickening argument. The approach aims to provide a canonical, atlas-free perspective analogous to moduli functors in algebraic geometry and discusses extending the framework to degenerations using log smooth manifolds. Together, these ideas lay groundwork for a virtual fundamental geometrical theory in derived and log-smooth settings, potentially unifying approaches to enumerative problems in symplectic and complex geometry.

Abstract

We summarize current work aimed at showing that moduli spaces of solutions to non-linear elliptic Fredholm partial differential equations are derived log smooth manifolds.
Paper Structure (7 sections, 5 theorems, 6 equations)

This paper contains 7 sections, 5 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Linear elliptic equations
  4. Derived smooth manifolds
  5. Moduli stacks of pseudo-holomorphic maps
  6. Representability
  7. Log smooth manifolds

Key Result

Theorem 3.6

For any perfect topological $\infty$-site $\mathsf E$ with finite limits, the $\infty$-category of topological functors $\mathsf{Der}\to\mathsf E$ preserving finite limits is equivalent, via restriction, to the $\infty$-category of topological functors $\mathsf{Sm}\to\mathsf E$ preserving finite pro

Theorems & Definitions (20)

  • Definition 3.1: Topological $\infty$-site
  • Example 3.2
  • Definition 3.3: Topological functor
  • Example 3.4
  • Definition 3.5: Derived smooth manifold
  • Theorem 3.6: Universal property of derived smooth manifolds
  • Lemma 3.7
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • ...and 10 more