April 30, 2024
Line Bundles on Complex Tori (Part 5)
Posted by John Baez
The Eisenstein integers are the complex numbers of the form where and are integers and . They form a subring of the complex numbers and also a lattice:
Last time I explained how the space of hermitian matrices is secretly 4-dimensional Minkowski spacetime, while the subset
is 3-dimensional hyperbolic space. Thus, the set of hermitian matrices with Eisenstein integer entries forms a lattice in Minkowski spacetime, and I conjectured that consists exactly of the hexagon centers in the hexagonal tiling honeycomb — a highly symmetrical structure in hyperbolic space, discovered by Coxeter, which looks like this:
Now Greg Egan and I will prove that conjecture.
April 26, 2024
Line Bundles on Complex Tori (Part 4)
Posted by John Baez
Last time I introduced a 2-dimensional complex variety called the Eisenstein surface
where is the lattice of Eisenstein integers. We worked out the Néron–Severi group of this surface: that is, the group of equivalence classes of holomorphic line bundles on this surface, where we count two as equivalent if they’re isomorphic as topological line bundles. And we got a nice answer:
where consists of hermitian matrices with Eisenstein integers as entries.
Now we’ll see how this is related to the ‘hexagonal tiling honeycomb’:
We’ll see an explicit bijection between so-called ‘principal polarizations’ of the Eisenstein surface and the centers of hexagons in this picture! We won’t prove it works — I hope to do that later. But we’ll get everything set up.
April 25, 2024
Line Bundles on Complex Tori (Part 3)
Posted by John Baez
You thought this series was dead. But it was only dormant!
In Part 1, I explained how the classification of holomorphic line bundles on a complex torus breaks into two parts:
the ‘discrete part’: their underlying topological line bundles are classified by elements of a free abelian group called the Néron–Severi group .
the ‘continuous part’: the holomorphic line bundles with a given underlying topological line bundle are classified by elements of a complex torus called the Jacobian .
In Part 2, I explained duality for complex tori, which is a spinoff of duality for complex vector spaces. I used this to give several concrete descriptions of the Néron–Severi group .
But the fun for me lies in the examples. Today let’s actually compute a Néron–Severi group and begin seeing how it leads to this remarkable picture by Roice Nelson:
This is joint work with James Dolan.
April 23, 2024
Moving On From Kent
Posted by David Corfield
Was it really seventeen years ago that John broke the news on this blog that I had finally landed a permanent academic job? That was a long wait – I’d had twelve years of temporary contracts after receiving my PhD.
And now it has been decided that I am to move on from the University of Kent. The University is struggling financially and has decreed that a number of programs, including Philosophy, are to be cut. Whatever the wisdom of their plan, my time here comes to an end this July.
What next? It’s a little early for me to retire. If anyone has suggestions, I’d be happy to hear them.
We started this blog just one year before I started at Kent. To help think things over, in the coming weeks I thought I’d revisit some themes developed here over the years to see how they panned out:
- Higher geometry: categorifying the Erlanger program
- Category theory meets machine learning
- Duality
- Categorifying logic
- Category theory applied to philosophy
- Rationality of (mathematical and scientific) theory change as understood through historical development
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 may seem merely cute, but they’re connected to some important ideas in algebra. To start seeing this, note that rescaling any Pythagorean triple gives a point with rational coordinates on the unit circle:
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 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 .
It’s true! And since is 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 -function of an elliptic curve, and you were supposed to shudder in horror. In this definition the -function is a product over all primes . 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 : 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 .
Now I’ll give a much better definition of the -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
where 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 -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 -functions. But there’s also a million-dollar prize for proving the Birch-Swinnerton–Dyer conjecture about -functions of elliptic curves. So if you want to learn about this stuff, you may try to learn the definition of an -function of an elliptic curve.
But in many expository accounts you’ll meet a big roadblock to understanding.
The -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 -function has this complicated appearance. Others say that tweaks must be made for bad primes to make sure the -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 and , which I’ll explain below. Monstrous Moonshine says that when is prime, the curve has genus zero iff divides the order of the Monster group, namely
Just for fun we’ve been looking at , among other cases. We used dessins d’enfant to draw a picture of , which seems to have genus , so for to have genus zero it seems we want the picture for 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.