## November 28, 2012

### Almost All of the First 50 Billion Groups Have Order 1024

#### Posted by Tom Leinster

Here’s an incredible fact: of the 50 billion or so groups of order at most 2000, more than 99% have order 1024. This was announced here:

Hans Ulrich Besche, Bettina Eick, E.A. O’Brien, The groups of order at most 2000.

Electronic Research Announcements of the American Mathematical Society7 (2001), 1–4.

By no coincidence, the paper was submitted in the year 2000. The real
advance was not that they had got up to order 2000, but that they had ‘developed
*practical* algorithms to construct or enumerate the groups of a given
order’.

I learned this amazing nugget from a recent MathOverflow answer of Ben Fairbairn.

You probably recognized that $1024 = 2^{10}$. A finite group is called a
‘$2$-group’ if the order of every element is a power of 2, or equivalently if the
order of the group is a power of 2. So as Ben points out, what this computation
suggests is that *almost every finite group is a 2-group*.

Does anyone know whether there are general results making this precise? Specifically, is it true that

$\frac{\text{number of 2-groups of order } \leq N}{\text{number of groups of order } \leq N} \to 1$

as $N \to \infty$?

## November 19, 2012

*TwoVect*

#### Posted by Urs Schreiber

**guest post by Jamie Vicary**

There are lots of computations in higher linear algebra that can be
difficult to carry out; not because any of the individual steps are
difficult, but because the calculation as a whole is long and
introduces many opportunities to make mistakes. A student of mine, Dan Roberts, wrote
an impressive computer package called *TwoVect* as an
add-on to *Mathematica* to help with these calculations, and I’d
like to tell you about it!

It’s already being used in earnest by Chris Douglas to find some new modular tensor categories, and by me and Dan for some projects in topological quantum field theory and quantum information. So we’re hopeful that the early bugs have been worked out, and we’re happy to show it to the world.

If you like semisimple monoidal categories, and ever spend time checking pentagon or hexagon equations, or computing the value of a string diagram, or verifying the axioms for a Frobenius algebra or a Hopf algebra, or checking properties like modularity or rigidity, or showing that your extended topological quantum field theory satisfies the right axioms, or checking the equations for a bimodule between two tensor categories — or you like the idea of using Mathematica’s built-in solvers to help with these tasks — or you just want to know what this stuff is all about! — read on.

## November 9, 2012

### Freedom From Logic

#### Posted by Mike Shulman

One of the most interesting things being discussed at IAS this year is the idea of developing a language for *informal* homotopy type theory. What does that mean? Well, traditional mathematics is usually written in natural language (with some additional helpful symbols), but in a way that all mathematicians can nevertheless recognize as “sufficiently rigorous” — and it’s generally understood that anyone willing to undertake the tedium *could* fully formalize it in a formal system like material set theory, structural set theory, or extensional type theory. By analogy, therefore, we would like an “informal” way to write mathematics in natural language which we can all agree *could* be fully formalized in homotopy type theory, by anyone willing to undertake the tedium.

### Back in Business

#### Posted by David Corfield

Sorry about that interruption in service. We seem to be back in business, but I fear we may have lost some of the more recent comments.

## November 1, 2012

### Parametrized Mates and Multivariable Adjunctions

#### Posted by Tom Leinster

*Guest post by Emily Riehl*

*(Note: the Café went down for a few days in early November 2012, and when Jacques got it back up again, some of the comments had been lost. I’ve tried to recreate them manually from my records, but I might have got some of the threading wrong.)*

The mates correspondence provides a means to transfer information from a diagram involving left adjoints to a diagram involving right adjoints. The basic observation is that in the presence of functors and adjunctions, arranged in the manner displayed below, there is a bijective correspondence between natural transformations $\lambda$ and $\rho$. $\array{ \mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' \\ {}^F\downarrow\dashv\uparrow^U& &{}^{F'}\downarrow\dashv\uparrow^{U'} \\ \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' }$

$\array{
\mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' & &
\mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' \\
{}^F \downarrow & {}^{\lambda}\swarrow & \downarrow^{F'} & &
{}^U\uparrow &\searrow^{\rho} & \uparrow^{U'} \\
\mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' & &
\mathcal{B} &\stackrel{B}{\to} & \mathcal{B}'
}$
Corresponding $\lambda$ and $\rho$ are called **mates** and are related by the pasting diagrams displayed here. Today I’d like to report on the preprint Multivariable adjunctions and mates, joint with Eugenia Cheng and Nick Gurski, which extends this notion to adjunctions with parameters. Our main theorem, also described in these slides describes the categorical structure that precisely captures the “naturality” of the **parametrized mates correspondence**.

But first let’s acquaint ourselves with some examples of ordinary mates.