---
date: 2022-09-18 21:38
modification date: Sunday 18th September 2022 21:38:24
title: "2022-09-18"
aliases: [2022-09-18]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #todo/untagged
- Refs:
	- <https://arxiv.org/pdf/math/9910179.pdf#page=3&zoom=160,-136,789>
- Links:
	- #todo/create-links 

---

# 2022-09-18


**Problem**: classify Lagrangian tori in $\PP^3\slice \CC$ up to Hamiltonian isotopy using any tools you have!

![](attachments/Pasted%20image%2020220919215709.png)
![](attachments/Pasted%20image%2020220919215938.png)

Main characterization of $\Fuk$: setting $\cat{C}(L, L') = \CF(L, L')$ with differential $\mu_1$, composition $\mu_2$, and higher operations $\mu_k$ makes $\Fuk(M, \omega)$ into 
- $\Lambda\dash$linear 
- $\ZZ\dash$graded 
- non-unital 
	- but cohomologically unital 
- $A_\infty$ category.

![](attachments/Pasted%20image%2020220918222958.png)
![](attachments/Pasted%20image%2020220918223039.png)
![](attachments/Pasted%20image%2020220918223124.png)

## Review

Goal: the higher products:
![](attachments/Pasted%20image%2020220918213840.png)
Motivation:
![](attachments/Pasted%20image%2020220918222129.png)

and e.g. in the DGA setting, the Massey product $\gens{x,y,z}_3 = m_3(x,y,z)$.
The moduli space appearing:
![](attachments/Pasted%20image%2020220918213930.png)
![](attachments/Pasted%20image%2020220918214001.png)

What's going on in pictures:
![](attachments/Pasted%20image%2020220918214036.png)
Compactify, look at boundary/ends:
![](attachments/Pasted%20image%2020220918214120.png)

$A_\infty$ relations:
![](attachments/Pasted%20image%2020220918214230.png)

Idea for $\Fuk$:
![](attachments/Pasted%20image%2020220918214301.png)



![](attachments/Pasted%20image%2020220918214351.png)

How to recover ordinary categories, but lose higher product information:
![](attachments/Pasted%20image%2020220918214433.png)

>  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

![](attachments/Pasted%20image%2020220918215951.png)

### Motivation from loop spaces

![](attachments/Pasted%20image%2020220918220152.png)
![](attachments/Pasted%20image%2020220918220203.png)
![](attachments/Pasted%20image%2020220918220211.png)
![](attachments/Pasted%20image%2020220918220241.png)

![](attachments/Pasted%20image%2020220918220546.png)
![](attachments/Pasted%20image%2020220918220606.png)
![](attachments/Pasted%20image%2020220918220723.png)
![](attachments/Pasted%20image%2020220918220900.png)

![](attachments/Pasted%20image%2020220918221150.png)
![](attachments/Pasted%20image%2020220918221855.png)


# Brass Tacks

![](attachments/Pasted%20image%2020220918222047.png)
![](attachments/Pasted%20image%2020220918222058.png)
where $\square$ is as above.

![](attachments/Pasted%20image%2020220918222250.png)

The cohomological category:
![](attachments/Pasted%20image%2020220918222402.png)

Being unital:
![](attachments/Pasted%20image%2020220918222415.png)

![](attachments/Pasted%20image%2020220918222456.png)

Equivalence:
![](attachments/Pasted%20image%2020220918222513.png)

![](attachments/Pasted%20image%2020220918222600.png)

# From PL book

![](attachments/Pasted%20image%2020220919220222.png)
![](attachments/Pasted%20image%2020220919220211.png)

![](attachments/Pasted%20image%2020220919220328.png)

![](attachments/Pasted%20image%2020220919220341.png)

![](attachments/Pasted%20image%2020220919220501.png)

![](attachments/Pasted%20image%2020220919220803.png)

![](attachments/Pasted%20image%2020220919220847.png)
![](attachments/Pasted%20image%2020220919220858.png)

![](attachments/Pasted%20image%2020220919220950.png)

![](attachments/Pasted%20image%2020220919221249.png)
![](attachments/Pasted%20image%2020220919221314.png)

![](attachments/Pasted%20image%2020220919221415.png)
![](attachments/Pasted%20image%2020220919221427.png)

![](attachments/Pasted%20image%2020220919221458.png)
![](attachments/Pasted%20image%2020220919221510.png)
![](attachments/Pasted%20image%2020220919221532.png)
![](attachments/Pasted%20image%2020220919221622.png)
![](attachments/Pasted%20image%2020220919221811.png)

![](attachments/Pasted%20image%2020220919221828.png)