Fukaya Category
Description of a certain wrapped Fukaya category \({\mathcal{O}}\): take the objects to be (Lagrangian) embedded curves, the morphisms are the graded abelian groups \(\hom_{\mathcal{O}}\coloneqq\qty{\bigoplus_{L_0 \pitchfork L_1} {\mathbf{Z}}/2{\mathbf{Z}}, {{\partial}}}\) where \({{\partial}}\) is given by counting holomorphic strips, localize along small isotopies.
Notes from Eisenbud
Add to Algebra qual review doc #todo
An ideal \({\mathfrak{p}}\) is prime iff \(JK \subset {\mathfrak{p}}\implies J \subset {\mathfrak{p}}\) or \(K\subset {\mathfrak{p}}\).
A ring is a domain iff the ideal \((0)\) is prime.
Inductively, if \({\mathfrak{p}}\) contains a product of ideals then it contains one of them.
Maximal ideals are prime, since \({\mathfrak{m}}\) maximal implies that \(R/{\mathfrak{m}}\) is a field.
A ring is local iff it has a unique maximal ideal \({\mathfrak{m}}\).
An element \(e\) is idempotent iff \(e^2 = e\).
An \(R{\hbox{}}\)algebra \(S\) is a ring \(S\) and a homomorphism \(\alpha:R \to S\).
Every ring is a \({\mathbf{Z}}{\hbox{}}\)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, \({\mathbf{Z}}\), or the localization of a ring at a prime ideal.
Random
 Steenbrink spectral sequence (PetersSteenbrink for exposition)
 RapoportZink spectral sequence
 Bounding ranks of curves over a function field: see elliptic fibrations
 Burnside ring (algebraic geometry) in AG: Take the free abelian group on finitely generated field extensions over a base field.
 Check statement of the BaezDolan cobordism hypothesis
Milnor K Theory in the Wild
See Milnor K theory

An appearance of Milnor \(K_2\) in the wild:
How Milnor Ktheory shows up in number theory: a conjecture by Tate and Birch:
Modular forms and DeligneSerre theorem

modular form yield 2dimensional Galois representations, and there is a classification theorem:
DeligneSerre Theorem:
The representation ring
Tags: #personal/idlethoughts

The representation ring \(R(G)\): the free \({\mathbf{Z}}{\hbox{}}\)module on isomorphism classes of irreducible representations.
 How can we construct this using modern groupoid yoga? Take the category \({}_{G}{\mathsf{Mod}}\), somehow restrict to just irreducible representations. 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 and then the geometric realization and then \(\pi_0\) or something? And then take the free \({\mathbf{Z}}{\hbox{}}\)module. I definitely need to ask some homotopy theorists how this construction goes for usual Ktheory in modern terms. So like… \begin{align*} {\mathbf{Z}}\left[ \pi_0 {\left\lvert { N \mathsf{C} } \right\rvert} \right] .\end{align*} 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 \(\mathsf{C}\). So maybe this needs to be something more simplicial set.