## June 18, 2024

### Magnitude Homology Equivalence

#### Posted by Tom Leinster

My brilliant and wonderful PhD student Adrián Doña Mateo and I have just arXived a new paper:

Adrián Doña Mateo and Tom Leinster. Magnitude homology equivalence of Euclidean sets. arXiv:2406.11722, 2024.

I’ve given talks on this work before, but I’m delighted it’s now in print.

Our paper tackles the question:

When do two metric spaces have the same magnitude homology?

We give an explicit, concrete, geometric answer for closed subsets of $\mathbb{R}^N$:

Exactly when their cores are isometric.

What’s a “core”? Let me explain…

## June 14, 2024

### 100 Papers on Magnitude

#### Posted by Tom Leinster

A milestone! By my count, there are now 100 papers on magnitude, including several theses, by a total of 73 authors. You can find them all at the magnitude bibliography.

Here I’ll quickly do two things: tell you about some of the hotspots of current activity, then — more importantly — describe several regions of magnitude-world that haven’t received the attention they could have, and where there might even be some low-hanging fruit.

## June 4, 2024

### 3d Rotations and the 7d Cross Product (Part 2)

#### Posted by John Baez

On Mathstodon, Paul Schwahn raised a fascinating question connected to the octonions. Can we explicitly describe an irreducible representation of $SO(3)$ on 7d space that preserves the 7d cross product?

I explained this question here:

This led to an intense conversation involving Layra Idarani, Greg Egan, and Paul Schwahn himself. The result was a shocking new formula for the 7d cross product in terms of the 3d cross product.

Let me summarize.

## May 27, 2024

### Lanthanides and the Exceptional Lie Group G_{2}

#### Posted by John Baez

The lanthanides are the 14 elements defined by the fact that their electrons fill up, one by one, the 14 orbitals in the so-called f subshell. Here they are:

lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium.

They are also called ‘rare earths’, but that term is often also applied to 3 more elements. Why? That’s a fascinating puzzle in its own right. But what matters to me now is something else: an apparent connection between the lanthanides and the exceptional Lie group G_{2}!

Alas, this connection remains profoundly mysterious to me, so I’m pleading for your help.

## May 26, 2024

### Wild Knots are Wildly Difficult to Classify

#### Posted by John Baez

In the real world, the rope in a knot has some nonzero thickness. In math, knots are made of infinitely thin stuff. This allows mathematical knots to be tied in infinitely complicated ways — ways that are impossible for knots with nonzero thickness! These are called ‘wild’ knots.

Check out the wild knot in this video by Henry Segerman. There’s just one point where it needs to have zero thickness. So we say it’s wild at just one point. But some knots are wild at many points.

There are even knots that are wild at *every* point! To build these you need to recursively put in wildness at more and more places, forever. I would like to see a good picture of such an everywhere wild knot. I haven’t seen one.

Wild knots are extremely hard to classify. This is not just a feeling — it’s a theorem. Vadim Kulikov showed that wild knots are harder to classify than any sort of countable structure that you can describe using first-order classical logic with just countably many symbols!

Very roughly speaking, this means wild knots are so complicated that we can’t classify them using anything we can write down. This makes them very different from ‘tame’ knots: knots that aren’t wild. Yeah, tame knots are hard to classify, but nowhere near *that* hard.

## May 15, 2024

### 3d Rotations and the 7d Cross Product (Part 1)

#### Posted by John Baez

There’s a dot product and cross product of vectors in 3 dimensions. But there’s also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There’s nothing really like this in other dimensions.

The following stuff is well-known: the group of linear transformations of $\mathbb{R}^n$ preserving the dot and cross product is called $SO(3)$. It consists of rotations. We say $SO(3)$ has an ‘irreducible representation’ on $\mathbb{R}^3$ because there’s no linear subspace of $\mathbb{R}^3$ that’s mapped to itself by every transformation in $SO(3)$, except for $\{0\}$ and the whole space.

Ho hum. But here’s something more surprising: it seems that $SO(3)$ also has an irreducible representation on $\mathbb{R}^7$ where every transformation preserves the dot product and cross product in 7 dimensions!

That’s right—no typo there. There is *not* an irreducible representation of $SO(7)$ on $\mathbb{R}^7$ that preserves the dot product and cross product. Preserving the dot product is easy. But the cross product in 7 dimensions is a strange thing that breaks rotation symmetry.

There *is*, apparently, an irreducible representation of the much smaller group $SO(3)$ on $\mathbb{R}^7$ that preserves the dot and cross product. But I only know this because people say Dynkin proved it! More technically, it seems Dynkin said there’s an $SO(3)$ subgroup of $G_2$ for which the irreducible representation of $\mathrm{G}_2$ on $\mathbb{R}^7$ remains irreducible when restricted to this subgroup. I want to see one explicitly.

## April 30, 2024

### Line Bundles on Complex Tori (Part 5)

#### Posted by John Baez

The Eisenstein integers $\mathbb{E}$ are the complex numbers of the form $a + b \omega$ where $a$ and $b$ are integers and $\omega = \exp(2 \pi i/3)$. They form a subring of the complex numbers and also a lattice:

Last time I explained how the space $\mathfrak{h}_2(\mathbb{C})$ of $2 \times 2$ hermitian matrices is secretly 4-dimensional Minkowski spacetime, while the subset

$\mathcal{H} = \left\{A \in \h_2(\mathbb{C}) \, \vert \, \det A = 1, \, \mathrm{tr}(A) \gt 0 \right\}$

is 3-dimensional hyperbolic space. Thus, the set $\mathfrak{h}_2(\mathbb{E})$ of $2 \times 2$ hermitian matrices with Eisenstein integer entries forms a lattice in Minkowski spacetime, and I conjectured that $\mathfrak{h}_2(\mathbb{E}) \cap \mathcal{H}$ 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**

$E = \mathbb{C}/\mathbb{E} \times \mathbb{C}/\mathbb{E}$

where $\mathbb{E} \subset \mathbb{C}$ is the lattice of Eisenstein integers. We worked out the Néron–Severi group $\mathrm{NS}(E)$ 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:

$\mathrm{NS}(E) \cong \mathfrak{h}_2(\mathbb{E})$

where $\mathfrak{h}_2(\mathbb{E})$ consists of $2 \times 2$ 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 $X$ 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 $\mathrm{NS}(X)$.

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 $\mathrm{Jac}(X)$.

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 $NS(X)$.

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 $3^2 + 4^2 = 5^2$ 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.