Fredholm topology and enumerative geometry: reflections on some words of Michael Atiyah

TL;DR

This paper surveys developments in differential and algebraic geometry related to enumerative questions and nonlinear elliptic PDEs from the 1980s, and explores potential extensions to 4-manifolds and complex structures on surfaces of general type.

Contribution

It provides a reflective overview of historical progress and proposes new directions for extending these ideas to complex structures on 4-manifolds.

Findings

Survey of 1980s developments in geometry and PDEs

Discussion of potential extensions to 4-manifolds

Reflections on Atiyah's influence and future research avenues

Abstract

The first part of this article is a short and selective survey of developments in differential and algebraic geometry from the 1980's involving enumerative questions and nonlinear elliptic partial differential equations. In the second part we discuss possible extensions of these ideas to structures on 4-manifolds, in particular to complex structures on surfaces of general type