---
date: 2021-05-01
tags: [ web/quick-notes ]
---


# 2021-05-01

## Notes on [Arithmetic Statistics](Arithmetic%20Statistics)

- Interesting question in arithmetic statistics: for $G \in \Grp$ finite, how many Galois extensions are there $K/\QQ$ with $G = \Gal(K/\QQ)$ and $\Delta \leq N$ ([discriminants](Unsorted/discriminants.md) for some fixed $N$?

- Example
	- For $G=\ZZ/2$, it is $O(N)$.
	- For $G=\ZZ/3$, it is $O(\sqrt N )$.

- One can ask a similar question about $\Cl(K)$ for $G\in \Ab$, or replacing $\QQ$ with [Unsorted/function field](Unsorted/function%20field.md) $\FF_q(t)$ for $q=p^n$, and ask questions about frequency of primes [ramified primes](ramified%20primes), [split primes](split%20primes), or remaining [inert primes](inert%20primes).

- Cool fact: there is an [equivalence of categories](equivalence%20of%20categories.md) between finitely-generated extensions $K/k$ with $\trdeg(K/k) = 1$ and regular projective [curves](curves.md) $C_{/k}$.
	- The (reverse) functor is the one sending a curve $C$ to its function field $k(C)$.

- [Hurwitz spaces](Hurwitz%20spaces) come up here!

![](attachments/image_2021-05-01-17-28-01.png)

- [etale cohomology](Unsorted/etale.md): 

![](attachments/image_2021-05-01-17-31-16.png)

![](attachments/image_2021-05-01-17-32-42.png)

- $\eps$ is a $q^i$ [Weil number](Weil%20number) if $\abs{ \iota(\eps) } = q^{i/2}$ for any embedding $\iota: \Qbar \embeds \CC$.

  - Examples: eigenvalues of geometric [Frobenius](Frobenius) acting on $H^i_c$.

![](attachments/image_2021-05-01-17-36-23.png)

![](attachments/image_2021-05-01-17-37-12.png)

- As a general philosophy, one should expect that [moduli space](moduli%20space.md) problems whose objects have nontrivial automorphisms are representable by [Unsorted/stacks MOC](Unsorted/stacks%20MOC.md), and those without nontrivial automorphisms are [representable](representable) by [scheme](scheme.md). 

- [Weil Conjectures Talks](Weil%20Conjectures%20Talks.md)

![](attachments/image_2021-05-01-17-48-07.png)

- [Homological Stability Course Notes](Homological%20Stability%20Course%20Notes.md) for [Hurwitz spaces](Hurwitz%20spaces) :

![](attachments/image_2021-05-01-17-51-07.png)

## Old Notes: Erik Schreyer

> Some old notes from March 10th, 2020

I talked to [Erik Schreyer](https://erikschreyer.wordpress.com/) today about some of the research he did with his advisor [Jason Cantarella](http://www.jasoncantarella.com/wordpress/), including his dissertation work (which he spoke about in the Geometry seminar last week) and a few other papers.

His dissertation work involved a cool way to represent arbitrary planar curves by *piecewise circular* arcs:

![](attachments/image-20200310232110171.png)

From what I understand, this involves fixing a curve (blue), choosing a collection of circles $C_1, \cdots C_n$ (black) such that each $C_i$ intersects $C_{i+1}$ in at least one distinguished point $p_i$ (pink). The curve traced out by following an arc on $C_i$ and switching to circle $C_{i+1}$ at $p_i$ is intended to yield a good approximation to the original curve, with certain regularity conditions at the $p_i$ (such as the first derivatives along both arcs agreeing at the point).

Erik's work actually seems to go a bit farther -- he has an algorithm (a *curve-closing operator*) that actually takes an *open* curve and produces a closed curve that is nearby in the $C_1$ norm. He uses this to construct piecewise circular approximations that consist of circles of *equal* radii, along with some control over the $C^1$ distance between the original curve and the approximation.

We also talked a bit about another problem Jason was working on, discussed in the following papers:

- [The symplectic geometry of closed equilateral random walks in 3-space (Cantarella, Shonkwiler 2013)](https://arxiv.org/abs/1310.5924)

- [A Fast Direct Sampling Algorithm for Equilateral Closed Polygons (Cantarella et al 2015)](https://arxiv.org/abs/1510.02466)

## Dirichlet's Theorem

Dirichlet's Theorem: An arithmetic progress with $(a, p) = 1$ contains infinitely many primes.
As a corollary, one can always find a *prime* $q$ that generates $\ZZ_p\units$ for any prime $p$.

## A SES isomorphic to a direct sum that does not split

> [Reference](http://math.stackexchange.com/questions/1082283/example-of-a-non-splitting-exact-sequence-0-%E2%86%92-m-%E2%86%92-m-oplus-n-%E2%86%92-n-%E2%86%92-0/1082313#1082313) 

Not every sequence of the form $0\to A \to A \oplus C \to C \to 0$ splits; take
$$
0 \to \ZZ \to \ZZ \oplus \bigoplus_\NN \ZZ/(2) \to \bigoplus_\NN \ZZ/(2) \to 0
$$
where the first map is multiplication by 2, the second is the quotient map and a right-shift. This can't split because $(1, 0, \cdots)$ has order 2 in the RHS but pulls back to $(1, 0) \oplus (2\ZZ \oplus 0)$ which has no element of order 2.

## Cogroups

See [cogroup](cogroup.md).