The n-Category Café

guest post by Christoph Wockel

Dear all:

We cordially invite you to participate in the workshop Higher Structures in Topology and Geometry V , which will take place May 25-27 in Hamburg (Germany). For information about speakers and location, you may consult our webpage:

• Higher Structures in Topology and Geometry V.

Best wishes,

Christoph Wockel (on behalf of the organisers Christoph Schweigert, Giorgio Trentinaglia and Chenchang Zhu)

It’s been more than a month since the last post in this thread… so let me remind you of the story so far.

I showed you how real and quaternionic quantum mechanics are lurking inside good old, ordinary complex quantum mechanics. Ordinary physicists only like complex Hilbert spaces. For example, they describe elementary particles using irreducible unitary representations of groups. And these are representations on complex Hilbert spaces. They couldn’t care less about real or quaternionic Hilbert spaces.

But — lo and behold! — the math gods have decreed that irreducible unitary representations come in three kinds:

• The real ones. These are the ones that you get by complexifying representations on real Hilbert spaces. These guys are isomorphic to their own dual. In fact, any representation $H$ of this kind comes with an invariant nondegenerate bilinear function $g: H \times H \to \mathbb{C}$ with $$g(v,w) = g(w,v) .$$
• The quaternionic ones. These are the ones that you get by taking representations on quaternionic Hilbert spaces, and thinking of them as complex Hilbert spaces. These guys are also isomorphic to their own dual. In fact, any representation $H$ of this kind comes with an invariant nondegenerate bilinear map $g: H \times H \to \mathbb{C}$ with $$g(v,w) = -g(w,v) .$$
• The truly complex ones. These are the ones that aren’t isomorphic to their own dual.

So, even if we don’t want to think about real and quaternionic Hilbert spaces, the math gods are hinting that we should!

And indeed, I showed you that an electron is a quaternion.

Well, that’s just a deliberately overdramatic way of putting it. More precisely, I showed you that any irreducible unitary representation of SU(2) is either real or quaternionic. And the famous ‘spin-1/2’ representation, which we use to describe electrons, is quaternionic. We usually describe the state of a spin-1/2 particle using two complex numbers. But we could also use a single quaternion!

Now, you might think this is a completely trivial statement, since you can take two complex numbers and combine them into one quaternion. That’s part of the idea — but what I’m saying is not quite that trivial. What I’m saying is that $SU(2)$ acts as quaternion-linear transformations that preserve the inner product on a 1-dimensional quaternionic Hilbert space — and that’s the spin-1/2 representation in disguise.

In fact all the half-integer-spin or ‘fermionic’ representations of SU(2) are quaternionic, while all its integer-spin or ‘bosonic’ representations are real.

I also showed you that a unitary group representation is quaternionic if and only if it comes equipped with an antiunitary operator $$J : H \to H$$ that commutes with all the group operations and obeys $$J^2 = -1 .$$ That’s not surprising: this operator is just multiplication by the quaternion $j$. But the cool part is that a unitary group representation is real if and only if it has a $J$ with all the same properties, except now $$J^2 = 1.$$ Now this operator comes from complex conjugation!

What’s the physical meaning of this operator $J$? For representations of $SU(2)$, it’s time reversal: it reverses the angular momentum!

Okay, now you’re caught up.

guest post by Tim Silverman

Welcome, welcome to the antepenultimate part of our series illustrating modular curves and their tilings. Last time, we talked about how the curves $X_0(N)$ are contructed out of their tiles, and illustrated how we could label the edges of the tiles, before going on to talk about how the labels of the edges could help understand how the edges glued together. But we only did this in a rather ad hoc way, and only for a couple of simple cases. We also talked about calculating the genus of modular curves.

This time, we’ll look at how, at least for prime $N$, we can work out, systematically, how the edges are glued together in more complicated cases; and we’ll start to look at the genus in particular cases. We’ll also revisit the elliptic points of these curves in slightly greater generality than last time.

I’m sinking in a sea of administrative duties at the moment, so for a bit of sanity I thought I’d jot down the glimmer of a thought I had.

We spoke back here about the term model for a set of ground terms, $X$, and a set of term constructors, $\Sigma$, as the initial algebra for the functor $F: Y \to X + \Sigma(Y)$. For example, the set of finite sequences of 1s is the term algebra for $F: Y \to 1 + Y$. More interesting examples involve trees.

Dually, there are coterms, which correspond to the behaviours of a system which unpacks an element into a term-constructor and collection of elements. These form the terminal coalgebra for a functor. So in the case of $F: Y \to 1 + Y$, coterms will be possibly infinite streams of 1s, each of which is unpacked, if not empty, into a 1 and another stream. Elements correspond to the extended natural numbers, i.e., the natural numbers with the infinite stream adjoined. In the case of trees, we find potentially infinitely deep trees, such as the trace trees of computations.

Now, the new thought. Tangles are like higher-dimensional pieces of syntax, as shown by their use, for example, in calculating with representations of quantum groups. This works through their forming the free braided monoidal category on a dualizable object. So, question: could there not be dually a system of ‘coterm’ tangles with possibly infinitely many subtangles?

One small hint that there’s something to this: a nice example of an initial algebra/terminal coalgebra pair was given by Tom here. It’s the dyadic rationals in the interval $[0, 1]$, and the real interval $[0, 1]$, as devised by Peter Freyd. There’s a completion process going on.

Now there’s a link from rationals to tangles in the form of Conway’s rational tangles, week 228 and week 229, where each two-stranded tangle may be assigned a rational in an isotopy-invariant way. Is there a ‘completion’? Well, Louis Kauffman and Sofia Lambropoulou tell us in On the classification of rational tangles that

each non-rational real number (algebraic or transcendental) can be associated to an infinite tangle…all the approximants of which are rational tangles. (p. 24)

There it is. Now I really must get back to my admin.

Maybe mathematicians are finally getting serious about the problems with journals:

• Workshop on Mathematics Journals, February 14-16, 2011, MSRI, Berkeley, California. Organized by James M Crowley (Society for Industrial and Applied Mathematics), Susan Hezlet (London Mathematical Society), Robion C Kirby (University of California, Berkeley), and Donald E McClure (American Mathematical Society).

Recall that predicative mathematics is mathematics which rejects “impredicative definitions,” and especially power sets. (Power sets are “impredicative” because you can say things like “let $X$ be the intersection of all blah subsets of $S$” and proceed to prove that $X$ is itself blah; thus $X$ has been defined “impredicatively” in terms of a collection of which it is a member.) So it might seem odd to claim that topos theory can make you a predicativist, since the basic ingredient in the definition of an elementary topos is a power object.

However, I mean instead to refer to Grothendieck topos theory. This is usually regarded as a sub-field of elementary topos theory, since every Grothendieck topos is an elementary topos. But actually, that’s really only true if $Set$ is itself an elementary topos! So if we were predicativists, we would instead say that $Set$ is some kind of pretopos, and we would conclude instead that any category of sheaves is a pretopos of the same kind. Moreover, we could do lots of ordinary Grothendieck-topos-theory with these “Grothendieck pretopoi” (modulo the usual sorts of difficulties that come up in doing any sort of mathematics predicatively). So “Grothendieck topos theory” is, aside from the confusion of names, fully compatible with predicativism.

But how can Grothendieck topos theory make you a predicativist?

Minhyong Kim announces that the Seminar on higher-dimensional algebra is beginning in London this week. One of Minhyong’s students, James Haydon, is giving the first of these – N-categories according to Baez and Dolan – on Thursday, 27, January, 2011 in KLB M204 of UCL.

In other news, Lieven le Bruyn tells us about a one-stop shop for mathematical blogs – Mathblogging.org.

Also while at Lieven’s blog, I noticed the guest post On the Reality of Noncommutative Space by Fred Van Oystaeyen, the author of Virtual Topology and Functorial Geometry (Taylor and Francis, 2009). The post ends with intriguing remark

In the book I mentioned how “free will” could be a noncommutative space aspect of the brain activity. I also mention a possible relations with string theory. I am not a specialist in all these things but now I reached the point that I “feel” noncommutative space is a better approximation of the reality and one should investigate it further.

I have a student starting a PhD with me on free will. Little does he know he’d better start learning functorial geometry.

This year’s major category theory meeting, Category Theory 2011, will take place in Vancouver on 17–23 July. The invited speakers have just been announced:

• Marcelo Aguiar (Texas A&M)
• Richard Garner (Macquarie)
• André Joyal (UQAM)
• Bill Lawvere (SUNY at Buffalo)
• Sandra Mantovani (Milan)
• Bob Paré (Dalhousie)

Get in the mood by browsing the conference website.

guest post by Tim Silverman

Welcome to the next part of our series The Big Colouring Book of Modular Curves. The last few times I was looking at $\Gamma_1(N)$ and $X_1(N)$. In this part, I want to move on to $\Gamma_0(N)$ and $X_0(N)$.

It’s three-month-old news, but I’ve only just heard it—and since many patrons of the Café enjoy hearing about academics taking power out of the hands of commercial journal publishers, I’ll pass it on.

So: in October last year, the entire editorial board of the Journal of Group Theory resigned. It came into effect on 1 January. Here’s a brief announcement. Unusually, the dispute appears not to have been about price, but poor service.

I learned this from the latest newsletter of the London Mathematical Society (LMS). Here’s the relevant part:

Another issue concerned the Journal of Group Theory, currently published by de Gruyter. It is by now public knowledge that the entire editorial board has resigned, and Susan Hezlet, the LMS Publisher, invited Council to consider whether the LMS should get involved in some way. For reasons of commercial confidentiality, however, our conclusions must for the moment remain under wraps.

I guess this means that the journal will now be published, under a slightly different name, by the LMS. A pattern seems to be emerging: editorial boards fed up with their (commercial) publishers are provided with a new and much better home by the LMS, to the benefit of everyone. This is a truly valuable service that the LMS provides.

Update (November 2018): I’m maintaining a bibliography of publications on magnitude at www.maths.ed.ac.uk/~tl/magbib.

Mark Meckes and I have just arXived one paper each on the magnitude of metric spaces:

• Mark Meckes, Positive definite metric spaces
• Tom Leinster, The magnitude of metric spaces.

This blog has played an important part in the development of the theory. I tried to send Jacques an email thanking him for his role in that, but his spam-blocking software blocked it. So Jacques, if you’re reading: thanks!

It’s a tangled story: you can find online at least 8 papers, 10 blog posts and 12 talks related to this subject, and they really didn’t appear in logical order. The purpose of this post is simply to round up the available material.

I hope that Mark, or I, or both of us, will feel like writing something expository about our new papers—but that’s for another day.

[Jim Stasheff is asking me to forward the following to the blog..]

guest post by Jim Stasheff

Here is a pdf file that I prepared which contains a discussion of parallel transport from a/my own homotopy point of view:

Parallel transport, holonomy and all that – a homotopy point of view

guest post by Tim Silverman

Welcome back to our series A Glorious Technicolor Panorama Documentary of Modular Form Country.

We’ve been looking at the curves $X_1(N)$, which result from quotienting the hyperbolic plane (or complex upper half-plane) by the action of the subgroup $\Gamma_1(N)$ of $PSL(2, \mathbb{Z})$. We do this by first quotienting by $\Gamma(N)$, giving a “mod $N$” version of our surface, and then further quotienting out by the additive or “translation” subgroup of the remaining action.

Last time, and the time before, we looked at rolling up sectors of $X(N)$ into cones, pictorially representing the second quotient process. Today, I want to take advantage of the fact that several of the $X_1(N)$ for small $N$ have genus $0$, to try to represent everything smoothly on the surface of a sphere.

I should warn you that the pictures you’re about to see are to some extent “hand-sketched”. I didn’t really have guides to how to lay out the sizes and proportions of the variously-shaped tiles, unlike in the case of the Platonic solids or other regular tilings, or the conical rolled-up sectors. So there’s no guarantee that these images are even conformally faithful, let alone in some sense canonically “correct”. I’ve done my best to make them look nice and at least be combinatorially correct, but … well … you have been warned. Take them with a pinch of salt. Not actual size. Contents may have settled in transit. Batteries not included. Cape does not really confer ability to fly.

With that out of the way, let’s plunge right in!

I have never quite got my head around the relationship between entailment and implication. Of course, having written this short post I now have a better understanding of my misunderstandings, at least. I was wondering if anyone could shed any more light on my thoughts.

Here in the UK we assess the research capabilities of our university departments every so many years via the RAE. What’s new for the next assessment exercise in 2014 (renamed REF) is an emphasis on impact. We have already seen this idea introduced in research funding, where one must explain what effects our research may have outside of academia.

The trouble for philosophy is how to fit into a model that seems to have been drawn from other disciplines. In the natural sciences, you may have carried out some biochemical research leading to a pharmaceutical intervention for a disease. In the socal sciences, research on gangs or teenage pregnancies might lead to new legislation and policy changes. Both of these show clear impact, but what can we hope for from philosophy?

Impact must take place outside of academia, so that influences on non-traditional academic partners cannot be counted as such. In view of the fact that it is already out of the ordinary for philosophers to work with other academics, I think this is unfortunate. Imagine that I could tell a tale of my philosophical work inspiring the setting up of this blog, which in turn helped to bring about the flourishing of online research collaboration in mathematics, as far as I can see, this would not be able to count as impact.

You might think that dissemination to the public could count as impact, but even here one must have strong evidence that considerable effects have happened. The giving of public lectures, or the appearance on radio or television by itself does not suffice. Unless there happened to be, say, ranks of high school teachers who have changed their practices because of things said at the Café, again that’s another non-starter.

I’m eager to hear how UK mathematicians are coping with this challenge. Is any work by, say, Café hosts Tom Leinster and Simon Willerton to be included in their respective departments’ impact narratives?