January 31, 2016
Integral Octonions (Part 12)
Posted by John Baez
guest post by Tim Silverman
“Everything is simpler mod $p$.”
That is is the philosophy of the Mod People; and of all $p$, the simplest is 2. Washed in a bath of mod 2, that exotic object, the $\mathrm{E}_8$ lattice, dissolves into a modest orthogonal space, its Weyl group into an orthogonal group, its “large” $\mathrm{E}_8$ 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 $\mathrm{E}_8$ 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 $\mathbb{F}_2$ geometry; I made a little progress in the comments, but then got completely lost. But there is indeed an extensive $\mathbb{F}_2$ world in here, parallel to the Euclidean one. I have finally found the key to it in the following fact:
Large $\mathrm{E}_8$ lattices mod $2$ are just maximal flats in a $7$-dimensional quadric over $\mathbb{F}_2$.
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 $\mathbb{F}_2$ 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’ $\mathrm{E}_8$ made of integral octonions, with shortest vectors of length $1$, contains ‘standard’ sized $\mathrm{E}_8$ lattices with vectors of minimal length $\sqrt{2}$, and Wilson’s Leech lattice construction comes out the right size. The other is $\sqrt{2}$ times larger: a ‘standard’ $\mathrm{E}_8$ lattice contains “large” $\mathrm{E}_8$ lattices of minimal length $2$, but Wilson’s Leech lattice construction gives something $\sqrt{2}$ times too big. I’ve chosen the latter convention because I find it less confusing: reducing the standard $\mathrm{E}_8$ mod $2$ 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.
January 17, 2016
Thinking about Grothendieck
Posted by John Baez
Here’s a new piece:
- Barry Mazur, Thinking about Grothendieck, January 6, 2016.
It’s short. I’ll quote just enough to make you want to read more.
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:
- Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmüller Groups.
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.
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:
- John Baez, Brendan Fong and Blake S. Pollard, A compositional framework for open Markov processes.
And here’s the basic idea….
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.
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!
December 28, 2015
A Compositional Framework for Passive Linear Networks
Posted by John Baez
My main interest these days is ‘network theory’. This means slightly different things to different people, but for me it’s the application of category theory to complex systems made from interacting parts, which can often be drawn using diagrams that look like graphs with extra labels. My dream is to set up a new kind of mathematics, applicable to living systems from cells to ecosystems. But I’ve been starting with more humble networks, like electrical circuits.
Brendan Fong, at Oxford and U. Penn, has been really crucial in developing this approach to network theory. Here’s our first paper:
• John Baez and Brendan Fong, A compositional framework for passive linear networks.
While my paper with Jason Erbele studied signal flow diagrams, this one focuses on circuit diagrams. The two are different, but closely related.
Instead of trying to explain the connection, let me just talk about this paper with Brendan. There’s a lot in here, so I’ll only explain the main result. It’s all about ‘black boxing’: hiding the details of a circuit and only remembering its behavior as seen from outside. But it involves fun stuff about symplectic geometry, and cospans, and Dirichlet forms, and other things.
December 22, 2015
Operads and Phylogenetic Trees
Posted by John Baez
A few years ago, after hearing Susan Holmes speak about the mathematics of phylogenetic trees, I became interested in their connection to algebraic topology. I wrote an article about it:
- John Baez, Operads and the tree of life, Azimuth, 6 July 2011.
In trying to the make the ideas precise I recruited the help of Nina Otter, who was then a graduate student at ETH Zürich. She came to Riverside and we started to work together.
Now Nina Otter is a grad student at Oxford working on mathematical biology with Heather Harrington. I visited her last summer and we made more progress… but then she realized that our paper needed another big theorem, a result relating our topology on the space of phylogenetic trees to the topology described by Susan Holmes and her coauthors here:
- Louis J. Billera, Susan P. Holmes and Karen Vogtmann, Geometry of the space of phylogenetic trees, Advances in Applied Mathematics 27 (2001), 733–767.
It took another half year to finish things up. I could never have done this myself.
But now we’re done! Here’s our paper:
- John Baez and Nina Otter, Operads and phylogenetic trees.
Let me tell you the basic idea….
December 8, 2015
Research Fellowships at Macquarie University
Posted by Emily Riehl
A little bird tells me that Macquarie University is hiring (even more) category theorists!
Specifically, they are offering two-year research fellowship positions, details of which can be found here.
Macquarie University, which is in greater Sydney, is the home for Centre of Australian Category Theory. As I’m sure many $n$-Café readers can attest, it’s a wonderful place to visit, to live, and to work. Applications are due in January.
December 5, 2015
Globular
Posted by John Baez
guest post by Jamie Vicary
When you’re trying to prove something in a monoidal category, or a higher category, string diagrams are a really useful technique, especially when you’re trying to get an intuition for what you’re doing. But when it comes to writing up your results, the problems start to mount. For a complex proof, it’s hard to be sure your result is correct — a slip of the pen could lead to a false proof, and an error that’s hard to find. And writing up your results can be a huge time-sink, requiring weeks or months using a graphics package, all just for some nice pictures that tell you little about the correctness of the proof, and become useless if you decide to change your approach. Computers should be able help with all these things, in the way that proof assistants like Coq and Agda are allowing us to work with traditional syntactic proofs in a more sophisticated way.
The purpose of this post is to introduce Globular, a new proof assistant for working with higher-categorical proofs using string diagrams. It’s available at http://globular.science, with documentation on the nab. It’s web-based, so everything happens right in your browser: build formal proofs, visualize and step through them; keep your proofs private, share them with collaborators, or make them publicly available.
Before we get into the technical details, here’s a screenshot of Globular in action:
The main part of the screen shows a diagram, which in this case is 2-dimensional. It represents a composite 2-cell in a finitely-presented 2-category, with the blue and red regions representing objects, the lines representing 1-cells, and the vertices representing 2-cells. In fact, this 2d diagram is just an intermediate state of a 3d proof, through which we’re navigating with the ‘Slice’ controls in the top-right. The proof itself has been built up by composing the generators listed in the signature, down the left-hand side of the screen. (If you want to take a look at this proof yourself, you can go straight there; in the top-right, set ‘Project’ to 0, then increment the second ‘Slice’ counter to scroll through the proof.)
Globular has been developed so far in the Quantum Group in the Oxford Computer Science department, by Krzysztof Bar, Katherine Casey, Aleks Kissinger, Jamie Vicary and Caspar Wylie. We haven’t quite got around to it yet, but Globular will be open-source, and we’re really keen for people to get involved and help build the software — there’s a huge amount to do! If you want to help out, get in touch.
December 3, 2015
42
Posted by John Baez
In The Hitchhiker’s Guide to the Galaxy, the number 42 was revealed to be the Answer to the Ultimate Question of Life, the Universe, and Everything. But we never learned what the question was!
That’s what I’ll explain this Saturday in Montreal.
My talk is part of the Canadian Mathematical Society winter meeting. But it’s open to the public, and I’ll keep the math simple and fun. So, come along and bring your kids!
It’s from 6 to 7 pm on December 5th at the Hyatt Regency Montreal, at 1255 Jeanne-Mance. It’s in “Rooms Soprano A & B”, but I won’t be singing — they must have confused me with my cousin Joan.
November 11, 2015
Burritos for Category Theorists
Posted by John Baez
You’ve probably heard of Lawvere’s Hegelian taco. Now here is a paper that introduces the burrito to category theorists:
- Ed Morehouse, Burritos for the hungry mathematician.
The source of its versatility and popularity is revealed:
To wit, a burrito is just a strong monad in the symmetric monoidal category of food.
November 10, 2015
Weil, Venting
Posted by Tom Leinster
From the introduction to André Weil’s Basic Number Theory:
It will be pointed out to me that many important facts and valuable results about local fields can be proved in a fully algebraic context, without any use being made of local compacity, and can thus be shown to preserve their validity under far more general conditions. May I be allowed to suggest that I am not unaware of this circumstance, nor of the possibility of similarly extending the scope of even such global results as the theorem of Riemann–Roch? We are dealing here with mathematics, not theology. Some mathematicians may think they can gain full insight into God’s own way of viewing their favorite topic; to me, this has always seemed a fruitless and a frivolous approach. My intentions in this book are more modest. I have tried to show that, from the point of view which I have adopted, one could give a coherent treatment, logically and aesthetically satisfying, of the topics I was dealing with. I shall be amply rewarded if I am found to have been even moderately successful in this attempt.
I was young when I discovered by harsh experience that even mathematicians with crashingly comprehensive establishment credentials can be as defensive and prickly as anyone. I was older when (and I only speak of my personal tastes) I got bored of tales of Grothendieck-era mathematical Paris.
Nonetheless, I find the second half of Weil’s paragraph challenging. Is there a tendency, in category theory, to imagine that there’s such a thing as “God’s own way of viewing” a topic? I don’t think that approach is fruitless. Is it frivolous?
November 3, 2015
Cakes, Custard, Categories and Colbert
Posted by John Baez
As you probably know, Eugenia Cheng has written a book called Cakes, Custard and Category Theory: Easy Recipes for Understanding Complex Maths, which has gotten a lot of publicity. In the US it appeared under the title How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics, presumably because Americans are less familiar with category theory and custard (not to mention the peculiar British concept of “pudding”).
Tomorrow, Wednesday November 4th, Eugenia will appear on The Late Show with Stephen Colbert. There will also be another lesser-known guest who looks like this:
Apparently his name is Daniel Craig and he works on logic—he proved something called the Craig interpolation theorem. I hear he and Eugenia will have a duel from thirty paces to settle the question of the correct foundations of mathematics.
Anyway, it should be fun! If you think I’m making this all up, go here. She’s really going to be on that show.
October 21, 2015
When Not To Use ‘The’
Posted by David Corfield
I’ve revised A Note on ‘The’ and ‘The Structure of’ in Homotopy Type Theory, which we discussed a few months ago – The Structure of A.
As Mike said back then, trying to define ‘structure of ’ in HoTT is a form of ‘noodling around’, and I rather think that working up a definition of ‘the’ is more important. The claim in the note is that we should only form a term ‘The $A$’ for a type $A$, if we have established the contractibility of $A$. I claim that this makes sense of types which are singleton sets, as well as the application of ‘the’ in cases where category theorists see universal properties, such as ‘the product of…’.
Going down the $h$-levels, contractible propositions are true ones. I think it’s not too much of a stretch to see the ‘the’ of ‘the fact that $P$’ as an indication of the same principle.
But what of higher $h$-levels? Is it the case that we don’t, or shouldn’t, use ‘the’ with types which are non-contractible groupoids? One case that came to mind is with algebraic closures of fields. Although people do say ‘the algebraic closure of a field $F$’ since any two such are isomorphic, as André Henriques writes here, a warning is often felt necessary about the use of ‘the’ in that these isomorphisms are not canonical. Do people here also get a little nervous with ‘the universal cover of a space’? Perhaps intuitively one provides a little extra structure (map in or map out, say) which makes the isotropy trivial.
I was also wondering if we see traces of this phenomena in natural language, but I think the examples I’m coming up with (the way to hang a symmetrical painting, the left of a pair of identical socks) are better thought of as concerning the formation of terms in equivariant contexts (as at nLab: infinity-action), and the subject of a lengthy discussion a while ago on coloured balls.