## April 19, 2024

### The Modularity Theorem as a Bijection of Sets

#### Posted by John Baez

*guest post by Bruce Bartlett*

John has been making some great posts on counting points on elliptic curves (Part 1, Part 2, Part 3). So I thought I’d take the opportunity and float my understanding here of the Modularity Theorem for elliptic curves, which frames it as an explicit bijection between sets. To my knowledge, it is not stated exactly in this form in the literature. There are aspects of this that I don’t understand (the explicit isogeny); perhaps someone can assist.

## April 18, 2024

### The Quintic, the Icosahedron, and Elliptic Curves

#### Posted by John Baez

Old-timers here will remember the days when Bruce Bartlett and Urs Schreiber were regularly talking about 2-vector spaces and the like. Later I enjoyed conversations with Bruce and Greg Egan on quintics and the icosahedron. And now Bruce has come out with a great article linking those topics to elliptic curves!

- Bruce Bartlett, The quintic, the icosahedron, and ellliptic curves,
*Notices of the American Mathematical Society***71**(April 2024), 447–453.

It’s expository and fun to read.

## April 17, 2024

### Pythagorean Triples and the Projective Line

#### Posted by John Baez

Pythagorean triples like $3^2 + 4^2 = 5^2$ may seem merely cute, but they’re connected to some important ideas in algebra. To start seeing this, note that rescaling any Pythagorean triple $m^2 + n^2 = k^2$ gives a point with rational coordinates on the unit circle:

$(m/k)^2 + (n/k)^2 = 1$

Conversely any point with rational coordinates on the unit circle can be scaled up to get a Pythagorean triple.

Now, if you’re a topologist or differential geometer you’ll know the unit circle is isomorphic to the real projective line $\mathbb{R}\mathrm{P}^1$ as a topological space, and as a smooth manifold. You may even know they’re isomorphic as real algebraic varieties. But you may never have wondered whether the points with *rational* coordinates on the unit circle form a variety isomorphic to the *rational* projective line $\mathbb{Q}\mathrm{P}^1$.

It’s true! And since $\mathbb{Q}\mathrm{P}^1$ is $\mathbb{Q}$ plus a point at infinity, this means there’s a way to turn rational numbers into Pythagorean triples. Working this out explicitly, this gives a nice explicit way to get our hands on all Pythagorean triples. And as a side-benefit, we see that points with rational coordinates are *dense* in the unit circle.

## April 15, 2024

### Semi-Simplicial Types, Part II: The Main Results

#### Posted by Mike Shulman

*(Jointly written by Astra Kolomatskaia and Mike Shulman)*

This is part two of a three part series of expository posts on our paper Displayed Type Theory and Semi-Simplicial Types. In this part, we cover the main results of the paper.

## April 10, 2024

### Machine Learning Jobs for Category Theorists

#### Posted by John Baez

Former Tesla engineer George Morgan has started a company called Symbolica to improve machine learning using category theory.

When Musk and his AI head Andrej Karpathy didn’t listen to Morgan’s worry that current techniques in deep learning couldn’t “scale to infinity and solve all problems,” Morgan left Tesla and started Symbolica. The billionaire Vinod Khosla gave him $2 million to prove that ideas from category theory could help.

Khosla later said “He delivered that, very credibly. So we said, ‘Go hire the best people in this field of category theory.’ ” He says that while he still believes in OpenAI’s continued success building large language models, he is “relatively bullish” on Morgan’s idea and that it will be a “significant contribution” to AI if it works as expected. So he’s invested $30 million more.

## March 28, 2024

### Why Mathematics is Boring

#### Posted by John Baez

I’m writing a short article with some thoughts on how to write math papers, with a provocative title. It’s due very soon, so if you have any thoughts about this draft I’d like to hear them soon!

## March 23, 2024

### Counting Points on Elliptic Curves (Part 3)

#### Posted by John Baez

In Part 1 of this little series I showed you Wikipedia’s current definition of the $L$-function of an elliptic curve, and you were supposed to shudder in horror. In this definition the $L$-function is a product over all primes $p$. But what do we multiply in this product? There are 4 different cases, each with its own weird and unmotivated formula!

In Part 2 we studied the 4 cases. They correspond to 4 things that can happen when we look at our elliptic curve over the finite field $\mathbb{F}_{p}$: it can stay smooth, or it can become singular in 3 different ways. In each case we got a formula for number of points the resulting curve over the fields $\mathbb{F}_{p^k}$.

Now I’ll give a much better definition of the $L$-function of an elliptic curve. Using our work from last time, I’ll show that it’s equivalent to the horrible definition on Wikipedia. And eventually I may get up the nerve to improve the Wikipedia definition. Then future generations will wonder what I was complaining about.

## March 13, 2024

### Counting Points on Elliptic Curves (Part 2)

#### Posted by John Baez

Last time I explained three ways that good curves can go bad. We start with an equation like

$y^2 = P(x)$

where $P$ is a cubic with integer coefficients. This may define a perfectly nice smooth curve over the complex numbers — called an ‘elliptic curve’ — and yet when we look at its solutions in finite fields, the resulting curves over those finite fields may fail to be smooth. And they can do it in three ways.

Let’s look at examples.

## March 10, 2024

### Counting Points on Elliptic Curves (Part 1)

#### Posted by John Baez

You’ve probably heard that there are a lot of deep conjectures about $L$-functions. For example, there’s the Langlands program. And I guess the Riemann Hypothesis counts too, because the Riemann zeta function is the grand-daddy of all $L$-functions. But there’s also a million-dollar prize for proving the Birch-Swinnerton–Dyer conjecture about $L$-functions of elliptic curves. So if you want to learn about this stuff, you may try to learn the definition of an $L$-function of an elliptic curve.

But in many expository accounts you’ll meet a big roadblock to understanding.

The $L$-function of elliptic curve is often written as a product over primes. For most primes the factor in this product looks pretty unpleasant… but worse, for a certain finite set of ‘bad’ primes the factor looks completely different, in one of 3 different ways. Many authors don’t explain *why* the $L$-function has this complicated appearance. Others say that tweaks must be made for bad primes to make sure the $L$-function is a modular form, and leave it at that.

I don’t think it needs to be this way.

## March 9, 2024

### Semi-Simplicial Types, Part I: Motivation and History

#### Posted by Mike Shulman

*(Jointly written by Astra Kolomatskaia and Mike Shulman)*

This is part one of a three-part series of expository posts on our paper Displayed Type Theory and Semi-Simplicial Types. In this part, we motivate the problem of constructing SSTs and recap its history.

## March 3, 2024

### Modular Curves and Monstrous Moonshine

#### Posted by John Baez

Recently James Dolan and I have been playing around with modular curves — more specifically the curves $X_0(n)$ and $X^+_0(n)$, which I’ll explain below. Monstrous Moonshine says that when $p$ is prime, the curve $X^+_0(p)$ has genus zero iff $p$ divides the order of the Monster group, namely

$p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71$

Just for fun we’ve been looking at $n = 11$, among other cases. We used *dessins d’enfant* to draw a picture of $X_0(11)$, which seems to have genus $1$, so for $X^+_0(11)$ to have genus zero it seems we want the picture for $X_0(11)$ to have a visible two-fold symmetry. After all, the torus is a two-fold branched cover of the sphere, as shown by Greg Egan here:

But we’re not seeing that two-fold symmetry. So maybe we’re making some mistake!

Maybe you can help us, or maybe you’d just like a quick explanation of what we’re messing around with.

## February 20, 2024

### Spans and the Categorified Heisenberg Algebra

#### Posted by John Baez

I’m giving this talk at the category theory seminar at U. C. Riverside, as a kind of followup to one by Peter Samuelson on the same subject. My talk will not be recorded, but here are the slides:

Abstract.Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from ‘spans’, where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a ‘categorified’ Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious, at least to me. However, Jeffery Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans.

## February 14, 2024

### Cartesian versus Symmetric Monoidal

#### Posted by John Baez

James Dolan and Chris Grossack and I had a fun conversation on Monday. We came up some ideas loosely connected to things Chris and Todd Trimble have been working on… but also connected to the difference between classical and quantum information.

## February 4, 2024

### The Atom of Kirnberger

#### Posted by John Baez

The 12th root of 2 times the 7th root of 5 is

$1.333333192495\dots$

And since the numbers 5, 7, and 12 show up in scales, this weird fact has implications for music! It leads to a remarkable meta-meta-glitch in tuning systems. Let’s check it out.

## January 29, 2024

### Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions

#### Posted by Tom Leinster

*Guest post by Matthew di Meglio*

Recently, my PhD supervisor Chris Heunen and I uploaded a preprint to arXiv giving an axiomatic characterisation of the category $\mathbf{FCon}$ of finite-dimensional Hilbert spaces and linear contractions. I thought it might be nice to explain here in a less formal setting the story of how this article came to be, including some of the motivation, ideas, and challenges.