The n-Category Café

Tonight I read in Lior Pachter’s blog:

I’m a (50%) professor of mathematics and (50%) professor of molecular & cell biology at UC Berkeley. There have been plenty of days when I have spent the working hours with biologists and then gone off at night with some mathematicians. I mean that literally. I have had, of course, intimate friends among both biologists and mathematicians. I think it is through living among these groups and much more, I think, through moving regularly from one to the other and back again that I have become occupied with the problem that I’ve christened to myself as the ‘two cultures’. For constantly I feel that I am moving among two groups — comparable in intelligence, identical in race, not grossly different in social origin, earning about the same incomes, who have almost ceased to communicate at all, who in intellectual, moral and psychological climate have so little in common that instead of crossing the campus from Evans Hall to the Li Ka Shing building, I may as well have crossed an ocean.

I try not to become preoccupied with the two cultures problem, but this holiday season I have not been able to escape it. First there was a blog post by David Mumford, a professor emeritus of applied mathematics at Brown University, published on December 14th. For those readers of the blog who do not follow mathematics, it is relevant to what I am about to write that David Mumford won the Fields Medal in 1974 for his work in algebraic geometry, and afterwards launched another successful career as an applied mathematician, building on Ulf Grenader’s Pattern Theory and making significant contributions to vision research. A lot of his work is connected to neuroscience and therefore biology. Among his many awards are the MacArthur Fellowship, the Shaw Prize, the Wolf Prize and the National Medal of Science. David Mumford is not Joe Schmo.

It therefore came as a surprise to me to read his post titled “Can one explain schemes to biologists?” in which he describes the rejection by the journal Nature of an obituary he was asked to write. Now I have to say that I have heard of obituaries being retracted, but never of an obituary being rejected. The Mumford rejection is all the more disturbing because it happened after he was invited by Nature to write the obituary in the first place!

The obituary Mumford was asked to write was for Alexander Grothendieck, a leading and towering figure in 20th century.

This spring, I will be teaching an undergraduate-level category theory course: Category theory in context. It has two aims:

(i) To provide a thorough “Cambridge-style” introduction to the basic concepts of category theory: representability, (co)limits, adjunctions, and monads.

(ii) To revisit as many topics as possible from the typical undergraduate curriculum, using category theory as a guide to deeper understanding.

For example, when I was an undergraduate, I could never remember whether the axioms for a group action required the elements of the group to act via automorphisms. But after learning what might be called the first lemma in category theory — that functors preserve isomorphisms — I never worried about this point again.

I announced the call for papers a few weeks ago. Now here’s the announcement of the conference itself. You’ll see I’m contributing on homotopy type theory. Pro-category theory attenders welcome.

CFR: SoTFoM, SYMPOSIUM II ‘COMPETING FOUNDATIONS?’; INSTITUTE OF PHILOSOPHY, LONDON, 12-13 January 2015.

The organisers are delighted to announce a provisional programme and call for registration for the upcoming Symposium in the Foundations of Mathematics, to be held at the Institute of Philosophy in London on 12-13th January 2015. There will be an additional (free) affiliated talk by Benedict Eastaugh at the Institute on the 14th January. Sponsors: The Institute of Philosophy, Mind Association, British Logic Colloquium, Aristotelian Society, British Society for the Philosophy of Science, and Birkbeck College.

On a scale of 0 to 10, how much does the average citizen of the Republic of Elbonia trust the president?

You’re conducting a survey to find out, and you’ve calculated that in order to get the precision you want, you’re going to need a sample of 100 statistically independent individuals. Now you have to decide how to do this.

You could stand in the central square of the capital city and survey the next 100 people who walk by. But these opinions won’t be independent: probably politics in the capital isn’t representative of politics in Elbonia as a whole.

So you consider travelling to 100 different locations in the country and asking one Elbonian at each. But apart from anything else, this is far too expensive for you to do.

Maybe a compromise would be OK. You could go to 10 locations and ask… 20 people at each? 30? How many would you need in order to match the precision of 100 independent individuals — to have an “effective sample size” of 100?

The answer turns out to be closely connected to a quantity I’ve written about many times before: magnitude. Let me explain…

I’m delighted to announce that Qiaochu Yuan has joined us as a host of the $n$-Category Café. Qiaochu is a grad student at Berkeley, who you’ll quite likely already know from his blog Annoying Precision or from his very wide and energetic activity at MathOverflow.

Welcome, Qiaochu!

You’ve probably heard about The Imitation Game, a film about Alan Turing’s life starring Benedict Cumberbatch. Maybe you’ve seen it.

On Sunday, the Edinburgh Filmhouse hosted a special event, “The Maths Behind The Imitation Game”, organized by Edinburgh undergraduates and the department’s Mathematics Engagement Officer, Julia Collins. To my surprise, it was packed out, with a hundred or so people in the audience, and more queuing for returns. Someone did a great job on publicity.

The event consisted of three talks, followed by Q&A. I gave one of them. Later, Julia wrote a blog post on each talk — these are nicely-written, and say much more than I’m going to say here.

• John Longley from Informatics spoke on computability and abstract notions of computation. Julia’s write-up is here.

• I spoke on the legacy of Turing’s code-breaking work at Bletchley Park, comparing and contrasting things then and now. The notes for my talk are here, and Julia’s post on it is here.

• Jamie Davies, an expert on the formation of tissues in mammals, spoke on the aspect of Turing’s work that’s perhaps least familiar to pure mathematicians: pattern formation and morphogenesis. For instance, you once consisted of just a single pair of cells. How did something so simple know how to develop into something as complex as you? Julia’s post is here.

Take a bunch of equal-sized solid balls in 8 dimensions. Pick one… and then get as many others to touch it as you can.

You can get 240 balls to touch it — no more. And unlike in 3 dimensions, there’s no ‘wiggle room’: you’re forced into a specific arrangement of balls, apart from your ability to rotate the whole configuration.

You can continue packing balls so that each ball touches 240 others — and unlike in 3 dimensions, there’s no choice about how to do it: their centers are forced to lie at a lattice of points called the E₈ lattice.

If we pick a point in this lattice, it has 240 nearest neighbors. Let’s call these the first shell. It has 2160 second-nearest neighbors. Let’s call these the second shell.

And here’s what fascinates me now…

Over the past year I have become increasingly fascinated by set theory and logic. So this morning when I was meant to be preparing a talk, I instead found myself thinking about the Cantor–Schröder–Bernstein theorem.

Theorem (Cantor–Schröder–Bernstein). Let $A$ and $B$ be sets. If there exist injections $f \colon A \to B$ and $g \colon B \to A$, then $|A|=|B|$.

This is an incredibly powerful tool for proving that two sets have the same cardinality. (Exercise: use CSB to prove that $|\mathbb{N}|=|\mathbb{Q}|$ and that $|P(\mathbb{N})|=|\mathbb{R}|$.) Earlier this fall, I taught a proof of this result that I learned from Peter Johnstone’s Notes on logic and set theory. The question that’s distracting me is this: how categorical is this argument?

Just for fun, here’s a reformulation of a famous conjecture in terms of the cartesian closed structure on the category of graphs.

Hedetniemi’s conjecture says there’s no clever way of colouring a product of graphs. By “colouring”, I mean colouring the vertices in such a way that the ends of each edge get different colours.

Given graphs $X$ and $Y$, any colouring of $X$ gives rise to a colouring of $X \times Y$: just paint $(x, y)$ the same colour as $x$. Of course, the same is true with $X$ and $Y$ interchanged. Hedetniemi conjectured that if you want to colour $X \times Y$ with as few colours as possible, you can’t beat this method. That is, writing $\chi$ for the chromatic number of a graph — the smallest number of colours needed to colour it —

$$\chi(X \times Y) = \min \{ \chi(X), \chi(Y) \}.$$

This has been open since 1966.

I’ll explain how Hedetniemi’s conjecture is equivalent to the following statement about the cartesian closed category of graphs: for any graph $X$ and complete graph $K$, either $K^X$ or $K^{K^X}$ has a point.

The Leech lattice gives the densest packing of spheres in 24 dimensions. The exceptional Jordan algebra, consisting of matrices

$$\left( \begin{array}{ccc} a & X & Y \\ X^* & b & Z \\ Y^* & Z^* & c \end{array} \right)$$

where $a,b,c$ are real and $X,Y,Z$ are octonions, has dimension $3 + 24 = 27$. They’re both remarkable entities. If the mathematical universe is a harmonious place, they should be connected.

More precisely: we should be able to fit the Leech lattice into the exceptional Jordan algebra in a nice way. And Greg Egan has shown that we can!

In fact, we can fit the Leech lattice into the space of matrices like this

$$\left( \begin{array}{ccc} 0 & X & Y \\ X^* & 0 & Z \\ Y^* & Z^* & 0 \end{array} \right)$$

where $X, Y, Z$ are integral octonions. We can do it in only finitely many ways — but Egan showed we can do it in at least 244,035,421 ways. Of these, at least 17,280 are compatible with the product on the exceptional Jordan algebra, in a way that I will describe.

Big numbers! Amusingly, Egan found the first number through a quick calculation which I’ll describe here — but the second, prettier number through an exhaustive computer search.