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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 Society 7 (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 101024 = 2^{10}. A finite group is called a ‘22-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

number of 2-groups of order  ≤Nnumber of groups of order  ≤N→1 \frac{\text{number of 2-groups of order }   \leq N}{\text{number of groups of order }   \leq N} \to 1

as N→∞N \to \infty?

Posted at 8:47 PM UTC | Permalink | Followups (30)

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.

Continue reading “TwoVect” …
Posted at 8:48 PM UTC | Permalink | Followups (16)

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.

Continue reading “Freedom From Logic” …
Posted at 3:19 PM UTC | Permalink | Followups (71)

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.

Posted at 8:56 AM UTC | Permalink | Followups (5)

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 λ\lambda and ρ\rho. 𝒜 →A 𝒜′ F↓⊣↑ U F′↓⊣↑ U′ ℬ →B ℬ′ \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}' }

 

𝒜 →A 𝒜′ 𝒜 →A 𝒜′ F↓ λ↙ ↓ F′      U↑ ↘ ρ ↑ U′ ℬ →B ℬ′ ℬ →B 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 λ\lambda and ρ\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.

Continue reading “Parametrized Mates and Multivariable Adjunctions” …
Posted at 11:53 PM UTC | Permalink | Followups (14)