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2022-09-18
Problem: classify Lagrangian tori in ${\mathbf{P}}^3_{/ {{\mathbf{C}}}} $ up to Hamiltonian isotopy using any tools you have!
Main characterization of \({\mathsf{Fuk}}\): setting \(\mathsf{C}(L, L') = \operatorname{CF}(L, L')\) with differential \(\mu_1\), composition \(\mu_2\), and higher operations \(\mu_k\) makes \({\mathsf{Fuk}}(M, \omega)\) into - \(\Lambda{\hbox{-}}\)linear - \({\mathbf{Z}}{\hbox{-}}\)graded - non-unital - but cohomologically unital - \(A_\infty\) category.
Review
Goal: the higher products: 
Motivation: 
and e.g. in the DGA setting, the Massey product \(\left\langle{x,y,z}\right\rangle_3 = m_3(x,y,z)\). The moduli space appearing: 
What’s going on in pictures: 
Compactify, look at boundary/ends: 
\(A_\infty\) relations: 
Idea for \({\mathsf{Fuk}}\): 
How to recover ordinary categories, but lose higher product information: 
After all, Fukaya categories are a particular approach to packaging holomorphic curve counts in a way that is particularly amenable to homological algebra.
Used to cook up new invariants of symplectic manifolds.
A_infty algebras
Motivation from loop spaces
Brass Tacks
where \(\square\) is as above.
The cohomological category: 
Being unital: 
Equivalence: 
From PL book
