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January 31, 2016

Integral Octonions (Part 12)

Posted by John Baez

guest post by Tim Silverman

“Everything is simpler mod pp.”

That is is the philosophy of the Mod People; and of all pp, the simplest is 2. Washed in a bath of mod 2, that exotic object, the E 8\mathrm{E}_8 lattice, dissolves into a modest orthogonal space, its Weyl group into an orthogonal group, its “large” E 8\mathrm{E}_8 sublattices into some particularly nice subspaces, and the very Leech lattice itself shrinks into a few arrangements of points and lines that would not disgrace the pages of Euclid’s Elements. And when we have sufficiently examined these few bones that have fallen out of their matrix, we can lift them back up to Euclidean space in the most naive manner imaginable, and the full Leech springs out in all its glory like instant mashed potato.

What is this about? In earlier posts in this series, JB and Greg Egan have been calculating and exploring a lot of beautiful Euclidean geometry involving E 8\mathrm{E}_8 and the Leech lattice. Lately, a lot of Fano planes have been popping up in the constructions. Examining these, I thought I caught some glimpses of a more extensive 𝔽 2\mathbb{F}_2 geometry; I made a little progress in the comments, but then got completely lost. But there is indeed an extensive 𝔽 2\mathbb{F}_2 world in here, parallel to the Euclidean one. I have finally found the key to it in the following fact:

Large E 8\mathrm{E}_8 lattices mod 22 are just maximal flats in a 77-dimensional quadric over 𝔽 2\mathbb{F}_2.

I’ll spend the first half of the post explaining what that means, and the second half showing how everything else flows from it. We unfortunately bypass (or simply assume in passing) most of the pretty Euclidean geometry; but in exchange we get a smaller, simpler picture which makes a lot of calculations easier, and the 𝔽 2\mathbb{F}_2 world seems to lift very cleanly to the Euclidean world, though I haven’t actually proved this or explained why — maybe I shall leave that as an exercise for you, dear readers.

N.B. Just a quick note on scaling conventions before we start. There are two scaling conventions we could use. In one, a ‘shrunken’ E 8\mathrm{E}_8 made of integral octonions, with shortest vectors of length 11, contains ‘standard’ sized E 8\mathrm{E}_8 lattices with vectors of minimal length 2\sqrt{2}, and Wilson’s Leech lattice construction comes out the right size. The other is 2\sqrt{2} times larger: a ‘standard’ E 8\mathrm{E}_8 lattice contains “large” E 8\mathrm{E}_8 lattices of minimal length 22, but Wilson’s Leech lattice construction gives something 2\sqrt{2} times too big. I’ve chosen the latter convention because I find it less confusing: reducing the standard E 8\mathrm{E}_8 mod 22 is a well-known thing that people do, and all the Euclidean dot products come out as integers. But it’s as well to bear this in mind when relating this post to the earlier ones.

Posted at 1:42 AM UTC | Permalink | Followups (71)

January 17, 2016

Thinking about Grothendieck

Posted by John Baez

Here’s a new piece:

It’s short. I’ll quote just enough to make you want to read more.

Posted at 3:15 AM UTC | Permalink | Followups (5)

January 16, 2016

Homotopy of Operads and Grothendieck-Teichmüller Groups

Posted by John Baez

Benoit Fresse has finished a big two-volume book on operads, which you can now see on his website:

He writes:

The first aim of this book project is to give an overall reference, starting from scratch, on the application of methods of algebraic topology to operads. To be more specific, one of our main objectives is the development of a rational homotopy theory for operads. Most definitions, notably fundamental concepts of operad and homotopy theory, are carefully reviewed in order to make our account accessible to a broad readership, which should include graduate students, as well as researchers coming from the various fields of mathematics related to our main topics.

The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, we set up the background required by an in-depth study of this subject, and we make precise the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.

Posted at 1:00 AM UTC | Permalink | Followups (1)

January 12, 2016

A Compositional Framework for Markov Processes

Posted by John Baez

Last summer my students Brendan Fong and Blake Pollard visited me at the Centre for Quantum Technologies, and we figured out how to understand open continuous-time Markov chains! I think this is a nice step towards understanding the math of living systems.

Admittedly, it’s just a small first step. But I’m excited by this step, since Blake and I have been trying to get this stuff to work for a couple years, and it finally fell into place. And we think we know what to do next. Here’s our paper:

And here’s the basic idea….

Posted at 12:28 PM UTC | Permalink | Followups (2)

January 10, 2016

On Digital Mathematics and Drive-By Contributors

Posted by Mike Shulman

At the Joint Mathematics Meetings last week there was a special session entitled Mathematical Information in the Digital Age of Science (MIDAS), with talks about structures to organize, disseminate, and formalize mathematics using computers and the Internet. The organizer, Patrick Ion, had invited me to give a talk based on my experience with projects such as the nLab and the HoTT Coq and Book projects. I had a hard time deciding what the audience at the session would benefit most from hearing, and I ended up changing the talk around right up until the minute I stood up to give it. But people seemed to like it, so I thought I would post the final version of the slides:

Part of my difficulty was in trying to extract some coherent message that would be memorable and useful. What I ended up with was a call to embrace plurality: be it in software, organizational structure, project goals, contributor involvement, or even mathematical foundations.

Posted at 4:52 AM UTC | Permalink | Followups (6)

January 6, 2016

Decorated Cospans

Posted by John Baez

I recently blogged about a paper I wrote with Brendan Fong. It’s about electrical circuits made of ‘passive’ components, like resistors, inductors and capacitors. We showed these circuits are morphisms in a category. Moreover, there’s a functor sending each circuit to its ‘external behavior’: what it does, as seen by someone who can only measure voltages and currents at the terminals.

Our paper uses a formalism that Brendan developed here:

• Brendan Fong, Decorated cospans, Theory and Applications of Categories 30 (2015), 1096–1120.

Let me explain this formalism. You don’t need to be an electrical engineer to like this!

Posted at 12:39 PM UTC | Permalink | Followups (5)