Lagrangian Floer theory, from geometry to algebra and back again

TL;DR

This paper surveys the development of Floer theory in symplectic geometry, highlighting its algebraic structures, applications to classical conjectures, and connections to mirror symmetry through the Fukaya category.

Contribution

It provides a comprehensive overview of Floer theory's evolution from geometric origins to algebraic and categorical frameworks, emphasizing its role in modern symplectic geometry and mirror symmetry.

Findings

Floer theory addresses classical conjectures of Arnold.

The Fukaya category captures rich algebraic structures.

Family Floer cohomology advances local-to-global principles.

Abstract

We survey various aspects of Floer theory and its place in modern symplectic geometry, from its introduction to address classical conjectures of Arnold about Hamiltonian diffeomorphisms and Lagrangian submanifolds, to the rich algebraic structures captured by the Fukaya category, and finally to the idea, motivated by mirror symmetry, of a "geometry of Floer theory" centered around family Floer cohomology and local-to-global principles for Fukaya categories.