opinion

I feel very strongly that the direction that things like the nLab, Lurie’s /Higher Topos Theory/, and the Stacks project *is the wrong one*. Like Grothendieck’s work it is very hard to consistency check these things and they do tend to doorstopness when published. Errors are subtle and lost in a sea of TeX. Moreover I feel that we’re recapitulating the historical territory that was already encroached on by the story that started with Frege and whose current torchbearer is Chaitin. I get antsy when reading papers that are many pages of category theory without alternate metaphor demonstrations, as if somehow the magic language fairy is going to bop us on the heads with the pixie dust of understanding because now we’re doing things in categories and toposes instead of sets. 

Rucker sees gnarl. I agree with him. We don’t know what to make of gnarl yet because many people are waiting for their lucky break at the Forget Functor Lottery: if only I have the right abstractions I can solve everything: hypergeneralization is just as nonproductive as foundationalism, and Chaitin’s comments about randomness aside: there is froth, there is nature. 

I get hopeful when I think that future interface rigs to mathematics are going to be like the accountant’s interface to the Hitchhiker’s Guide to the Galaxy’s financial systems (see /Mostly Harmless/). 

Furthermore (and this isn’t my torch to bear), I think that the mathematics that is being taught children is the wrong sort: why teach them to be gimcrack calculating machines and later force them to unlearn commandments issued by clueless teachers when we could be using the articulation of the computer to give them intuitive understandings of concepts that have never been available before in history? And think of what kind of mathematicians those children could turn out to be when they grow up? It’s not fair to show kids fractals and then say “it’s time to do this algebra!” for two very important reasons: allowing the child to play with the former would allow them to have a very kinesthetic/visual emotional way about reasoning that would be more useful to developing mathematical competency and interest than dragging them through arbitrary (from the point of the historical contingency of mathematical symbolism) and capricious symbol manipulation rules for which they’re being judged on their ability to apply flawlessly, and it is unfair in the waving a carrot on a stick in front of the kid and then giving them an ant to eat when they have solved the problem to the satisfaction of idiotically backwards grading system. We’re actively reducing the pool size of potential brilliant mathematicians with the current setup. 

complete graph K37, laser etched.

With Ike Feitler‘s help:

The complete graph K_{37}, laser etched on acrylic. There
is an svg of the design at thingiverse, so if you have a laser etcher, you can make it yourself.

According to Wolfram Alpha, K_{37} is
arc-transitive, biconnected, bridgeless, cage, Cayley graph, chromatically unique, circulant, claw-free , complete, connected, cyclic, determined by spectrum, distance-regular, distance-transitive, edge-transitive, Hamilton-connected, Hamiltonian, Hamming, integral, Johnson, Kneser, LCF, line graph, nonplanar, perfect, regular, strongly regular, symmetric, traceable, Turán, unitransitive, and vertex-transitive. (links will come with a future edit to this post).

It has graph spectrum (-1)^{36}36^{1} and 666 (spooky!) edges. The graph stability index is 1.203 trillion.

arcs of the Riemann zeta function’s nontrivial roots

In Boston, we had some rainbows today:

But I want to share with you a different sort of rainbow arc.

I recently got Elias Wegert’s amazing book Visual Complex Functions for my 35th birthday. Along the edges of some of the pages are phase portraits of the critical strip of the Riemann zeta function \zeta(s). Elias has pictures of these strips up on his website. The yellow diagonals are fairly straightforward to see, and he’s written The Riemann Zeta Function on Arithmetic Progressions about them, in which we find out that there is a /stochastic period/ of order 2\pi/log(2).

I used gimp’s zealouscrop function and imagemagick to glue these strips together, yielding this picture:

See those arcs? We don’t know anything about them.

Matt McIrvin’s experiments with SAGE seem to suggest that the arcs are properties of the phase of the zeta function and aren’t immediately apparent if one just has a picture of the roots themselves. The Fourier Transform of Dirac deltas supported at the roots has led to some questions by John Baez:
Quasicrystals and the Riemann Hypothesis
and this post at the n-category cafe: Quasicrystals and the Riemann Hypothesis

As for the arcs, I think one logical stepping stone would be to try to see if other L-functions also have them, and there’s both SAGE’s implementation of the Dokchitser L-functions implementation and the LMFDB to poke around with.

product of ζ(n) over N\1 and random p-ary matrices.

Let’s take a look at:

\prod_{n=2}^{\infty} \zeta(n)

Where \zeta(n) is the Riemann zeta function of n. Numerically this is around
2.2948565916733137, but that’s not so interesting. What’s interesting is what happens when
we write out the Euler product for \zeta(n):

\prod_{n=2}^{\infty} \zeta(n) = \prod_{n=2}^{\infty} \prod_{p \mathrm{prime}} \frac{1}{1-p^{-n}}

\prod_{p \,\mathrm{prime}} \prod_{n=2}^{\infty}  \frac{1}{1-p^{-n}} = \prod_{p \,\mathrm{prime}} \left[\prod_{n=2}^{\infty} 1-\frac{1}{p^{n}}\right]^{-1}

So maybe we want this instead:

\prod_{n=2}^{\infty} \frac{1}{\zeta(n)} = \prod_{p \,\mathrm{prime}} \left[\prod_{n=2}^{\infty} 1-\frac{1}{p^{n}}\right] =

\prod_{p \,\mathrm{prime}} \frac{p}{p-1} \left[\prod_{n=1}^{\infty} 1-\frac{1}{p^{n}}\right]  \approx 0.43575707677264559373\ldots

What’s interesting about this is that \left[\prod_{n=1}^{\infty} 1-\frac{1}{2^{n}}\right] corresponds to oeis:A048651 — “This is the probability that a large random binary matrix is nonsingular ”, and it’s not hard to see that this generalizes to large random p-ary matrices.

Remarks:
1. There is some connection between the nontrivial zeros of \zeta(s) and random matrices, the Gaussian Unitary Ensemble (GUE) in particular. But it is interesting that we have gotten to random matrices without even considering the zeroes.
2. The product \left[\prod_{n=2}^{\infty} 1-\frac{1}{p^{n}}\right] can be expressed in terms of a Jacobian theta functions or a q-Pochhammer symbol. It is instructive to note that we have gotten to the territory of modular forms without having to deal with Hecke’s generalization of the identity Riemann used to construct the zeta function from theta functions.
3. The constant 0.435757076772645593\ldots is mentioned in Steven Finch‘s Errata and Addenda to Mathematical Constants on page 30 where he says of it “makes a small appearence (as a certain
best probability corresponding to finite nilpotent groups)”

The Bruhat-Tits tree of SL(2), Cubic Julia sets, and Thompson groups.

This morning I watch this video of James Arthur talking about a poster about the history of the Langlands programme:

I was struck by the shape of the Bruhat-Tits building in the background of the image. It’s topologically isotropic, which is a feature of (filled) Julia sets.

Bill Casselman wrote The Bruhat-Tits tree of SL(2) in which there are pictures of it.

Remark one: there is a topological embedding of the Bruhat-Tits tree of SL(2) into a cubic Julia set:

Remark two: there is a Thompson group which corresponds to the symmetries of the Bruhat-Tits tree of SL(2),
of which here is depicted one transformation thereof:

The following papers about the automorphisms of the Bruhat-Tits tree may be relevant:
Groups of hierarchomorphisms of trees and related
Hilbert spaces
by Yurii A. Neretin, though it is not clear at all whether the hierarchomorphisms he speaks of are akin to the symmetry operation visualized above, because the one visualized above requires no gluing or cutting at all. There are a few other hits on google for “”bruhat-tits tree” and “julia set””, but only 13 documents match total. Jim Belk’s paper A Thompson Group for the Basilica may also be relevant.