## May 22, 2015

### PROPs for Linear Systems

#### Posted by John Baez

PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about ‘signal-flow diagrams’ in control theory—an important branch of engineering. I talked about that here on the *n*-Café a while ago, but it’s time for an update.

### How to Acknowledge Your Funder

#### Posted by Tom Leinster

A comment today by Stefan Forcey points out ways in which US citizens can try to place legal limits on the surveillance powers of the National Security Agency, which we were discussing in the context of its links with the American Mathematical Society. If you want to act, time is of the essence!

But Stefan also tells us how he resolved a dilemma. Back here, he asked Café patrons what he should do about the fact that the NSA was offering him a grant (for non-classified work). Take their money and contribute to the normalization of the NSA’s presence within the math community, or refuse it and cause less mathematics to be created?

What he decided was to accept the funding and — in this paper at least — include a kind of protesting acknowledgement, citing his previous article for the *Notices of the AMS*.

I admire Stefan for openly discussing his dilemma, and I think there’s a lot to be said for how he’s handled it.

## May 21, 2015

### The Origin of the Word “Quandle”

#### Posted by John Baez

A quandle is a set equipped with a binary operation with number of properties, the most important being that it distributes over itself:

$a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c)$

They show up in knot theory, where they capture the essence of how the strands of a knot cross over each other… yet they manage to give an *invariant* of a knot, independent of the way you draw it. Even better, the quandle is a complete invariant of knots: if two knots have isomorphic quandles, there’s a diffeomorphism of $\mathbb{R}^3$ mapping one knot to the other.

I’ve always wondered where the name ‘quandle’ came from. So I decided to ask their inventor, David Joyce—who also proved the theorem I just mentioned.

## May 18, 2015

### The Revolution Will Not Be Formalized

#### Posted by Mike Shulman

After a discussion with Michael Harris over at the blog about his book Mathematics without apologies, I realized that there is a lot of confusion surrounding the relationship between homotopy type theory and computer formalization — and that moreover, this confusion may be causing people to react negatively to one or the other due to incorrect associations. There are good reasons to be confused, because the relationship is complicated, and various statements by prominent members of both communities about a “revolution” haven’t helped matters. This post and its sequel(s) are my attempt to clear things up.

## May 13, 2015

### Categorifying the Magnitude of a Graph

#### Posted by Simon Willerton

Tom Leinster introduced the idea of the magnitude of graphs (first at the Café and then in a paper). I’ve been working with my mathematical brother Richard Hepworth on categorifying this and our paper has just appeared on the arXiv.

Categorifying the magnitude of a graph, Richard Hepworth and Simon Willerton.

The magnitude of a graph can be thought of as an integer power series. For example, consider the Petersen graph.

Its magnitude starts in the following way. $\begin{aligned} \#P&=10-30q+30q^{2}+90q ^{3}-450q^{4}\\ &\quad\quad+810q^{5} + 270 q^{6} - 5670 q^{7} +\dots. \end{aligned}$

Richard observed that associated to each graph $G$ there is a bigraded group $\mathrm{MH}_{\ast ,\ast }(G)$, the **graph magnitude homology** of $G$, that has the graph magnitude $# G$ as its graded Euler characteristic.
$\begin{aligned}
#G &= \sum _{k,l\geqslant 0} (-1)^{k}\cdot \mathrm{rank}\bigl (\mathrm{MH}_{k,l}(G)\bigr )\cdot q^{l}\\
&= \sum _{l\geqslant 0} \chi \bigl (\mathrm{MH}_{\ast ,l}(G)\bigr )\cdot q^{l}.
\end{aligned}$
So graph magnitude homology categorifies graph magnitude in the same sense that Khovanov homology categorifies the Jones polynomial.

For instance, for the Petersen graph, the ranks of $\mathrm{MH}_{k,l}(P)$ are given in the following table. You can check that the alternating sum of each row gives a coefficient in the above power series.

$\begin{array}{rrrrrrrrrr} &&&&&&k\\ &&0&1&2&3&4&5&6&7 \\ &0 & 10\\ & 1 & & 30 \\ &2 & && 30 \\ &3 &&& 120 & 30 \\ l &4 &&&& 480 & 30 \\ &5 &&&&& 840 & 30 \\ &6 &&&&& 1440 & 1200 & 30 \\ &7 &&&&&& 7200 & 1560 & 30 \\ \\ \end{array}$

Many of the properties that Tom proved for the magnitude are shadows of properties of magnitude homology and I’ll describe them here.

## April 30, 2015

### Breakfast at the n-Category Café

#### Posted by Emily Riehl

Michael Harris recently joined us for breakfast at the *n*-category café. Perhaps some readers here would be interested in following the postprandial discussion that is underway on his blog: Mathematics Without Apologies.

## April 28, 2015

### Categories in Control

#### Posted by John Baez

To understand ecosystems, ultimately will be to understand networks.- B. C. Patten and M. Witkamp

A while back I decided one way to apply my math skills to help save the planet was to start pushing toward green mathematics: a kind of mathematics that can interact with biology and ecology just as fruitfully as traditional mathematics interacts with physics. As usual with math, the payoffs will come slowly, but they may be large. It’s not a substitute for doing other, more urgent things—but if mathematicians don’t do this, who will?

As a first step in this direction, I decided to study *networks*.

This May, a small group of mathematicians is meeting in Turin for a workshop on the categorical foundations of network theory, organized by Jacob Biamonte. I’m trying to get us mentally prepared for this. We all have different ideas, yet they should fit together somehow.

Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now I want to say a bit about what I’ve been doing with Jason Erbele.

## April 24, 2015

### A Synthetic Approach to Higher Equalities

#### Posted by Mike Shulman

At last, I have a complete draft of my chapter for Elaine Landry’s book *Categories for the working philosopher*. It’s currently titled

**Homotopy Type Theory: A synthetic approach to higher equalities**. pdf

As you can see (if you read it), not much is left of the one fragment of draft that I posted earlier; I decided to spend the available space on HoTT itself rather than detour into synthetic mathematics more generally. Although the conversations arising from that draft were still helpful, and my other recent ramblings did make it in.

Comments, questions, and suggestions would be very much appreciated! It’s due this Sunday (I got an extension from the original deadline), so there’s a very short window of time to make changes before I have to submit it. I expect I’ll be able to revise it again later in the process, though.

## April 12, 2015

### The Structure of A

#### Posted by David Corfield

I attended a workshop last week down in Bristol organised by James Ladyman and Stuart Presnell, as part of their Homotopy Type Theory project.

Urs was there, showing everyone his magical conjuring trick where the world emerges out of the opposition between **$\emptyset$** and **$\ast\;$** in Modern Physics formalized in Modal Homotopy Type Theory.

Jamie Vicary spoke on the Categorified Heisenberg Algebra. (See also John’s page.) After the talk, interesting connections were discussed with dependent linear type theory and tangent (infinity, 1)-toposes. It seems that André Joyal and colleagues are working on the latter. This should link up with Urs’s Quantization via Linear homotopy types at some stage.

As for me, I was speaking on the subject of my chapter for the book that Mike’s *Introduction to Synthetic Mathematics* and John’s *Concepts of Sameness* will appear in. It’s on reviving the philosophy of geometry through the (synthetic) approach of cohesion.

In the talk I mentioned the outcome of some further thinking about how to treat the phrase ‘the structure of $A$’ for a mathematical entity. It occurred to me to combine what I wrote in that discussion we once had on The covariance of coloured balls with the analysis of ‘the’ from The King of France thread. After the event I thought I’d write out a note explaining this point of view, and it can be found here. Thanks to Mike and Urs for suggestions and comments.

The long and the short of it is that there’s no great need for the word ‘structure’ when using homotopy type theory. If anyone has any thoughts, I’d like to hear them.

## April 7, 2015

### Information and Entropy in Biological Systems

#### Posted by John Baez

I’m helping run a workshop on Information and Entropy in Biological Systems at NIMBioS, the National Institute of Mathematical and Biological Synthesis, which is in Knoxville Tennessee.

I think you’ll be able to watch live streaming video of this workshop while it’s taking place from Wednesday April 8th to Friday April 10th. Later, videos will be made available in a permanent location.

To watch the workshop live, go **here**. Go down to where it says

Investigative Workshop: Information and Entropy in Biological Systems

Then click where it says **live link**. There’s nothing there *now*, but I’m hoping there will be when the show starts!

## April 6, 2015

### Five Quickies

#### Posted by Tom Leinster

I’m leaving tomorrow for an “investigative workshop” on Information and Entropy in Biological Systems, co-organized by our own John Baez, in Knoxville, Tennessee. I’m excited! And I’m hoping to learn a lot.

A quick linkdump before I go:

## March 14, 2015

### Split Octonions and the Rolling Ball

#### Posted by John Baez

You may enjoy these webpages:

because they explain a nice example of the Erlangen Program more tersely — and I hope more simply — than before, with the help of some animations made by Geoffrey Dixon using WebGL. You can actually get a ball to roll in way that illustrates the incidence geometry associated to the exceptional Lie group $\mathrm{G}_2$!

Abstract.Understanding exceptional Lie groups as the symmetry groups of more familiar objects is a fascinating challenge. The compact form of the smallest exceptional Lie group, $\mathrm{G}_2$, is the symmetry group of an 8-dimensional nonassociative algebra called the octonions. However, another form of this group arises as symmetries of a simple problem in classical mechanics! The space of configurations of a ball rolling on another ball without slipping or twisting defines a manifold where the tangent space of each point is equipped with a 2-dimensional subspace describing the allowed infinitesimal motions. Under certain special conditions, the split real form of $\mathrm{G}_2$ acts as symmetries. We can understand this using the quaternions together with an 8-dimensional algebra called the ‘split octonions’. The rolling ball picture makes the geometry associated to $\mathrm{G}_2$ quite vivid. This is joint work with James Dolan and John Huerta, with animations created by Geoffrey Dixon.

## March 11, 2015

### A Scale-Dependent Notion of Dimension for Metric Spaces (Part 1)

#### Posted by Simon Willerton

Consider the following shape that we are zooming into and out of. What dimension would you say it was?

At small scales (or when it is very far away) it appears as a point, so seems zero-dimensional. At bigger scales it appears to be a line, so seems one-dimensional. At even bigger scales it appears to have breadth as well as length, so seems two-dimensional. Then, finally, at very large scales it appears to be made from many widely separated points, so seems zero-dimensional again.

We arrive at an important observation:

The perceived dimension varies with the scale at which the shape is viewed.

Here is a graph of my attempt to capture this perceived notion of dimension mathematically.

Hopefully you can see that, moving from the left, you start off at zero, then move into a region where the function takes value around one, then briefly moves up to two then drops down to zero again.

The ‘shape’ that we are zooming into above is actually a grid of $3000\times 16$ points as that’s all my computer could handle easily in making the above picture. If I’d had a much bigger computer and used say $10^{7}\times 10^{2}$ points then I would have got something that more obviously had both a one-dimensional and two-dimensional regime.

In this post I want to explain how this idea of dimension comes about. It relies on a notion of ‘size’ of metric spaces, for instance, you could use Tom’s notion of magnitude, but the above picture uses my notion of $2$-spread.

I have mentioned this idea before at the Café in the post on my spread paper, but I wanted to expand on it somewhat.

## March 9, 2015

### HITs and the Erlangen Program

#### Posted by Mike Shulman

Last week I had a minor epiphany, which I would like to share. If it holds up, it’ll probably end up somewhere in my chapter for Landry’s book.

I was fortunate during my formal education to have the opportunity to take a fair number of physics classes in addition to mathematics. Over that time I was introduced to tensors (more specifically, tensor fields on manifolds) several times in a row: first in special relativity, then in differential geometry, and then again in general relativity and Riemannian geometry.

During all those classes, I remember noticing that mathematicians and physicists tended to speak about tensors in different ways. To a mathematician, a tensor (field) was a global geometric object associated to a manifold; upon choosing a local chart it could be expressed using local coordinates, but fundamentally it was a geometric thing. By contrast, physicists tended to talk about a tensor as a collection of coordinates labeled by indices, with its “tensorial” nature encoded in how those coordinates transformed upon changing coordinates.

(To be sure, most physicists nowadays are aware of the mathematician’s perspective, and many even use it. But I still got the sense that they found it easier, or at least expected that we students would find it easier, to think in coordinates.)

Here is my epiphany: the symbiosis between these two viewpoints on tensors is closely related to the symbiosis between HITs and univalence in homotopy type theory. I will now explain…

## March 5, 2015

### Mathematics Without Apologies

#### Posted by David Corfield

Since I had reviewed the manuscript for Princeton University Press eighteen months ago, this week I received my complementary copy of Mathematics Without Apologies by Michael Harris.

Michael, as most people here will know, is a number theorist whose successes include, with Richard Taylor, the proof of the local Langlands conjecture for the general linear group $GL_n(K)$ in characteristic 0. But alongside being a prize-winning mathematician, he also likes to think hard about the nature of mathematics. He spoke at a conference I co-organised, Two Streams in the Philosophy of Mathematics, and I’ve met up with him on a number of other occasions, including the Delphi meeting with John. He’s extremely well placed then to give an account of the life of a current mathematician with, as suggested by the book’s subtitle, *Portrait of a problematic vocation*, all its peculiarities.

I’ll be re-reading the book once term is over, and say more then, but for now those wanting to find out more can read a Q&A with the author, and drafts of some chapters available here. Michael has also set up an associated blog, named after the book.