Introduction to Fukaya Categories - James Pascaleff

Dive into the world of Fukaya categories with this comprehensive lecture, the first in a minicourse series. Explore the fundamental concepts of symplectic geometry essential for understanding Fukaya categories. Begin with an introduction to symplectic manifolds and Lagrangian submanifolds, then delve into exactness conditions, almost complex structures, and holomorphic maps. Examine Maslov indices and gradings, and gain insights into the historical context of classical mechanics. Investigate infinitesimal symplectic geometry in tangent spaces, local classifications, and exactness. Conclude with an exploration of exact Lagrangians, intersections, and the Lagrange multipliers picture. This lecture provides a solid foundation for further study in Floer cohomology and practical examples of Fukaya categories in subsequent sessions.

Intro
Outline
Symplectic structures
History: Classical Mechanics
Infinitesimal (tangent space) symplectic geometry
Infinitesimal (tangent space) Lagrangian submanifolds
Structures on the tangent bundle
Local classification
Exactness
Exact Lagrangians
Let's intersect
Lagrange multipliers picture