---
date: 2021-04-25
tags: [ web/quick-notes ]
---


# 2021-04-25

## Fukaya Category

Description of a certain wrapped [Fukaya category](Fukaya%20category.md) $\OO$: take the objects to be (Lagrangian) embedded curves, the morphisms are the graded abelian groups $\hom_\OO \definedas \qty{\bigoplus_{L_0 \transverse L_1} \ZZ/2\ZZ, \bd}$ where $\bd$ is given by counting holomorphic strips, localize along small isotopies.


## Notes from Eisenbud

Add to Algebra qual review doc #todo

An ideal $\mfp$ is prime iff $JK \subset \mfp \implies J \subset \mfp$ or $K\subset \mfp$.

A ring is a domain iff the ideal $(0)$ is prime.

> Inductively, if $\mfp$ contains a product of ideals then it contains one of them.

Maximal ideals are prime, since $\mm$ maximal implies that $R/\mm$ is a field.

A ring is *local* iff it has a unique maximal ideal $\mm$.

An element $e$ is idempotent iff $e^2 = e$.

An $R\dash$algebra $S$ is a ring $S$ and a homomorphism $\alpha:R \to S$.

Every ring is a $\ZZ\dash$algebra in a unique way.

The most interesting commutative algebras are $S/I$ where $S = k[x_1, \cdots, x_n]$ for $k$ a field, $\ZZ$, or the [localization](Unsorted/localization%20of%20rings.md) of a ring at a prime ideal.


## Random

- Steenbrink spectral sequence (Peters-Steenbrink for exposition)
- Rapoport-Zink spectral sequence
- Bounding ranks of curves over a [function field](Unsorted/function%20field.md): see [elliptic fibrations](elliptic%20fibrations)
- [Burnside ring (algebraic geometry)](Burnside%20ring%20(algebraic%20geometry).md) in AG: 
	Take the free abelian group on finitely generated field extensions over a base field.
- Check statement of the Baez-Dolan [cobordism hypothesis](cobordism%20hypothesis.md)



## Milnor K Theory in the Wild

See [Milnor K theory](Milnor%20K%20theory)

- An appearance of Milnor $K_2$ in the wild:

  How Milnor K-theory shows up in number theory: a conjecture by Tate and Birch: ![](attachments/image_2021-04-25-12-25-47.png)
 

## Modular forms and Deligne-Serre theorem


- [modular form](modular%20form.md) yield 2-dimensional [Galois representations](Galois%20representations.md), and there is a classification theorem:

  Deligne-Serre Theorem: ![](attachments/image_2021-04-25-12-43-06.png)


## The representation ring

Tags: #personal/idle-thoughts

- The [representation ring](representation%20ring.md) $R(G)$: the free \(\ZZ\dash\)module on isomorphism classes of irreducible [representations](Unsorted/Representation%20Theory%20(Subject%20MOC).md).

  - How can we construct this using modern [groupoid](groupoid.md) yoga? 
  Take the category \( \modsleft{G} \), somehow restrict to just [irreducible representations](irreducible%20representations).
  Maybe there's a better thing to do here though, like "ignoring" reducibles the same way John Carlson "ignored" projectives.
  But okay, anyway, take that category. 
  Take its [nerve](nerve.md) and then the [geometric realization](geometric%20realization.md) and then $\pi_0$ or something?
  And then take the free \(\ZZ\dash\)module.
  I definitely need to ask some homotopy theorists how this construction goes for usual [K-theory](Unsorted/K-theory.md) in modern terms.
  So like...
  \[
  \ZZ \left[ \pi_0 \abs{ N \cat{C} } \right]
  .\]
  The $\pi_0$ should be taking isomorphism classes somehow, but maybe this only works for groupoids?
  But okay, whatever, I just need a functor that takes categories into spaces where two objects end up in the same path component iff they're isomorphic in $\cat{C}$. 
  So maybe this needs to be something more [simplicial set](simplicial%20set.md).