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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.

Posted at 4:06 PM UTC | Permalink | Post a Comment

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!

It’s expository and fun to read.

Posted at 10:31 AM UTC | Permalink | Post a Comment

April 17, 2024

Pythagorean Triples and the Projective Line

Posted by John Baez

Pythagorean triples like 3 2+4 2=5 23^2 + 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 2m^2 + n^2 = k^2 gives a point with rational coordinates on the unit circle:

(m/k) 2+(n/k) 2=1(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 P 1\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 P 1\mathbb{Q}\mathrm{P}^1.

It’s true! And since P 1\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.

Posted at 9:52 AM UTC | Permalink | Followups (2)

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.

Posted at 2:41 AM UTC | Permalink | Post a Comment

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.

Posted at 4:29 PM UTC | Permalink | Followups (6)

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!

Posted at 10:21 PM UTC | Permalink | Followups (36)

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 LL-function of an elliptic curve, and you were supposed to shudder in horror. In this definition the LL-function is a product over all primes pp. 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 𝔽 p\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 𝔽 p k\mathbb{F}_{p^k}.

Now I’ll give a much better definition of the LL-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.

Posted at 1:00 AM UTC | Permalink | Followups (7)

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) y^2 = P(x)

where PP 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.

Posted at 8:00 PM UTC | Permalink | Followups (10)

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 LL-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 LL-functions. But there’s also a million-dollar prize for proving the Birch-Swinnerton–Dyer conjecture about LL-functions of elliptic curves. So if you want to learn about this stuff, you may try to learn the definition of an LL-function of an elliptic curve.

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

The LL-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 LL-function has this complicated appearance. Others say that tweaks must be made for bad primes to make sure the LL-function is a modular form, and leave it at that.

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

Posted at 9:08 PM UTC | Permalink | Followups (4)

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.

Posted at 5:33 PM UTC | Permalink | Followups (2)

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)X_0(n) and X 0 +(n)X^+_0(n), which I’ll explain below. Monstrous Moonshine says that when pp is prime, the curve X 0 +(p)X^+_0(p) has genus zero iff pp divides the order of the Monster group, namely

p=2,3,5,7,11,13,17,19,23,29,31,41,47,59,71 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=11n = 11, among other cases. We used dessins d’enfant to draw a picture of X 0(11)X_0(11), which seems to have genus 11, so for X 0 +(11)X^+_0(11) to have genus zero it seems we want the picture for X 0(11)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.

Posted at 9:43 PM UTC | Permalink | Followups (4)

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.

Posted at 10:51 PM UTC | Permalink | Followups (4)

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.

Posted at 6:46 AM UTC | Permalink | Followups (12)

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 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.

Posted at 8:09 PM UTC | Permalink | Followups (3)

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 FCon\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.

Posted at 1:31 PM UTC | Permalink | Followups (6)