July 27, 2016
Topological Crystals (Part 2)
Posted by John Baez
We’re building crystals, like diamonds, purely from topology. Last time I said how: you take a graph $X$ and embed its maximal abelian cover into the vector space $H_1(X,\mathbb{R})$.
Now let me back up and say a bit more about the maximal abelian cover. It’s not nearly as famous as the universal cover, but it’s very nice.
July 23, 2016
Topological Crystals (Part 1)
Posted by John Baez
Over on Azimuth I posted an article about crystals:
- John Baez, Diamonds and triamonds, Azimuth, 11 April 2016.
In the comments on that post, a bunch of us worked on some puzzles connected to ‘topological crystallography’—a subject that blends graph theory, topology and mathematical crystallography. You can learn more about that subject here:
- Tosio Sunada, Crystals that nature might miss creating, Notices of the AMS 55 (2008), 208–215.
I got so interested that I wrote this paper about it, with massive help from Greg Egan:
- John Baez, Topological crystals.
I’ll explain the basic ideas in a series of posts here.
June 11, 2016
How the Simplex is a Vector Space
Posted by Tom Leinster
It’s an underappreciated fact that the interior of every simplex $\Delta^n$ is a real vector space in a natural way. For instance, here’s the 2-simplex with twelve of its 1-dimensional linear subspaces drawn in:
(That’s just a sketch. See below for an accurate diagram by Greg Egan.)
In this post, I’ll explain what this vector space structure is and why everyone who’s ever taken a course on thermodynamics knows about it, at least partially, even if they don’t know they do.
May 26, 2016
Good News
Posted by John Baez
Various bits of good news concerning my former students Alissa Crans, Derek Wise, Jeffrey Morton and Chris Rogers.
May 20, 2016
Castles in the Air
Posted by Mike Shulman
The most recent issue of the Notices includes a review by Slava Gerovitch of a book by Amir Alexander called Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. As the reviewer presents it, one of the main points of the book is that science was advanced the most by the people who studied and worked with infinitesimals despite their apparent formal inconsistency. The following quote is from the end of the review:
If… maintaining the appearance of infallibility becomes more important than exploration of new ideas, mathematics loses its creative spirit and turns into a storage of theorems. Innovation often grows out of outlandish ideas, but to make them acceptable one needs a different cultural image of mathematics — not a perfectly polished pyramid of knowledge, but a freely growing tree with tangled branches.
The reviewer makes parallels to more recent situations such as quantum field theory and string theory, where the formal mathematical justification may be lacking but the physical theory is meaningful, fruitful, and made correct predictions, even for pure mathematics. However, I couldn’t help thinking of recent examples entirely within pure mathematics as well, and particularly in some fields of interest around here.
May 19, 2016
The HoTT Effect
Posted by David Corfield
Martin-Löf type theory has been around for years, as have category theory, topos theory and homotopy theory. Bundle them all together within the package of homotopy type theory, and philosophy suddenly takes a lot more interest.
If you’re looking for places to go to hear about this new interest, you are spoilt for choice:
- CFA: Foundations of Mathematical Structuralism, Munich, 12-14 October 2016 (see below for a call for papers).
- FOMUS, Foundations of Mathematics: Univalent foundations and set theory, Bielefeld, 18-23 July 2016.
- Homotopy Type Theory in Logic, Metaphysics and Philosophy of Physics, Bristol, 13-15 September 2016.
For an event which delves back also to pre-HoTT days, try my
- Type Theory and Philosophy, Canterbury, 9-10 June 2016.
May 12, 2016
E_{8} as the Symmetries of a PDE
Posted by John Huerta
My friend Dennis The recently gave a new description of the Lie algebra of $\mathrm{E}_8$ (as well as all the other complex simple Lie algebras, except $\mathfrak{sl}(2,\mathbb{C})$) as the symmetries of a system of partial differential equations. Even better, when he writes down his PDE explicitly, the exceptional Jordan algebra makes an appearance, as we will see.
- Dennis The, Exceptionally simple PDE.
This is a story with deep roots: it goes back to two very different models for the Lie algebra of $\mathrm{G}_2$, one due to Cartan and one due to Engel, which were published back-to-back in 1893. Dennis figured out how these two results are connected, and then generalized the whole story to nearly every simple Lie algebra, including $\mathrm{E}_8$.
May 10, 2016
The Works of Charles Ehresmann
Posted by John Baez
Charles Ehresmann’s complete works are now available for free here:
There are 630 pages on algebraic topology and differential geometry, 800 pages on local structures and ordered categories, and their applications to topology, 900 pages on structured categories and quotients and internal categories and fibrations, and 850 pages on sketches and completions and sketches and monoidal closed structures.
That’s 3180 pages!
On top of this, more issues of the journal he founded, Cahiers de Topologie et Géométrie Différentielle Catégoriques, will become freely available online.
May 8, 2016
Man Ejected from Flight for Solving Differential Equation
Posted by Tom Leinster
A professor of economics was escorted from an American Airlines flight and questioned by secret police after the woman in the next seat spotted him writing page after page of mysterious symbols. It’s all over the internet. Press reports do not specify which differential equation it was.
Although his suspiciously mediterranean appearance may have contributed to his neighbour’s paranoia, the professor has the privilege of not having an Arabic name and says he was treated with respect. He’s Italian. The flight was delayed by an hour or two, he was allowed to travel, and no harm seems to have been done.
Unfortunately, though, this story is part of a genre. It’s happening depressingly often in the US that Muslims (and occasionally others) are escorted off planes and treated like criminals on the most absurdly flimsy pretexts. Here’s a story where some passengers were afraid of the small white box carried by a fellow passenger. It turned out to contain baklava. Here’s one where a Berkeley student was removed from a flight for speaking Arabic, and another where a Somali woman was ejected because a flight attendant “did not feel comfortable” with her request to change seats. The phenomenon is now common enough that it has acquired a name: “Flying while Muslim”.
May 5, 2016
Which Paths Are Orthogonal to All Cycles?
Posted by John Baez
Greg Egan and I have been thinking about topological crystallography, and I bumped into a question about the homology of a graph embedded in a surface, which I feel someone should have already answered. Do you know about this?
I’ll start with some standard stuff. Suppose we have a directed graph $\Gamma$. I’ll write $e : v \to w \,$ when $e$ is an edge going from the vertex $v$ to the vertex $w$. We get a vector space of 0-chains $C_0(\Gamma,\mathbb{R})$, which are formal linear combinations of vertices, and a vector space of 1-chains $C_1(\Gamma,\mathbb{R})$, which are formal linear combinations of edges. We also get a boundary operator
$\partial : C_1(\Gamma,\mathbb{Z}) \to C_0(\Gamma,\mathbb{Z})$
sending each edge $e: v \to w$ to the difference $w - v$. A 1-cycle is 1-chain $c$ with $\partial c = 0$. There is an inner product on 1-chains for which the edges form an orthonormal basis.
Any path in the graph gives a 1-chain. When is this 1-chain orthogonal to all 1-cycles?
To make this precise, and interesting, I should say a bit more.
May 4, 2016
Categorifying Lucas’ Equation
Posted by John Baez
In 1875, Édouard Lucas challenged his readers to prove this:
A square pyramid of cannon balls contains a square number of cannon balls only when it has 24 cannon balls along its base.
In other words, the 24th square pyramid number is also a perfect square:
$1^2 + 2^2 + \cdots + 24^2 = 70^2$
and this is only true for 24. Nitpickers will note that it’s also true for 0 and 1. However, these are the only three natural numbers $n$ such that $1^2 + 2^2 + \cdots + n^2$ is a perfect square.
This fact was only proved in 1918, with the help of elliptic functions. Since then, more elementary proofs have been found. It may seem like much ado about nothing, but actually this fact about the number 24 underlies the simplest construction of the Leech lattice! So, understanding it better may be worthwhile.
Gavin Wraith has a new challenge, which is to find a bijective proof that the number 24 has this property. But part of this challenge is to give a precise statement of what counts as success!
I’ll let him explain…
April 30, 2016
Relative Endomorphisms
Posted by Qiaochu Yuan
Let $(M, \otimes)$ be a monoidal category and let $C$ be a left module category over $M$, with action map also denoted by $\otimes$. If $m \in M$ is a monoid and $c \in C$ is an object, then we can talk about an action of $m$ on $c$: it’s just a map
$\alpha : m \otimes c \to c$
satisfying the usual associativity and unit axioms. (The fact that all we need is an action of $M$ on $C$ to define an action of $m$ on $c$ is a cute instance of the microcosm principle.)
This is a very general definition of monoid acting on an object which includes, as special cases (at least if enough colimits exist),
- actions of monoids in $\text{Set}$ on objects in ordinary categories,
- actions of monoids in $\text{Vect}$ (that is, algebras) on objects in $\text{Vect}$-enriched categories,
- actions of monads (letting $M = \text{End}(C)$), and
- actions of operads (letting $C$ be a symmetric monoidal category and $M$ be the monoidal category of symmetric sequences under the composition product)
This definition can be used, among other things, to straightforwardly motivate the definition of a monad (as I did here): actions of a monoidal category $M$ on a category $C$ correspond to monoidal functors $M \to \text{End}(C)$, so every action in the above sense is equivalent to an action of a monad, namely the image of the monoid $m$ under such a monoidal functor. In other words, monads on $C$ are the universal monoids which act on objects $c \in C$ in the above sense.
Corresponding to this notion of action is a notion of endomorphism object. Say that the relative endomorphism object $\text{End}_M(c)$, if it exists, is the universal monoid in $M$ acting on $c$: that is, it’s a monoid acting on $c$, and the action of any other monoid on $c$ uniquely factors through it.
April 23, 2016
Polygonal Decompositions of Surfaces
Posted by John Baez
If you tell me you’re going to take a compact smooth 2-dimensional manifold and subdivide it into polygons, I know what you mean. You mean something like this picture by Norton Starr:
or this picture by Greg Egan:
(Click on the images for details.) But what’s the usual term for this concept, and the precise definition? I’m writing a paper that uses this concept, and I don’t want to spend my time proving basic stuff. I want to just refer to something.
April 21, 2016
Type Theory and Philosophy at Kent
Posted by David Corfield
I haven’t been around here much lately, but I would like to announce this workshop I’m running on 9-10 June, Type Theory and Philosophy. Following some of the links there will show, I hope, the scope of what may be possible.
One link is to the latest draft of an article I’m writing, Expressing ‘The Structure of’ in Homotopy Type Theory, which has evolved a little over the year since I posted The Structure of A.
March 31, 2016
Foundations of Mathematics
Posted by John Baez
Roux Cody recently posted an interesting article complaining about FOM — the foundations of mathematics mailing list:
- Roux Cody, on Foundations of Mathematics (mailing list), 29 March 2016.
Cody argued that type theory and especially homotopy type theory don’t get a fair hearing on this list, which focuses on traditional set-theoretic foundations.
This will come as no surprise to people who have posted about category-theoretic foundations on this list. But the discussion became more interesting when Harvey Friedman, the person Cody was implicitly complaining about, joined in. Friedman is a famous logician who posts frequently on Foundations of Mathematics. He explained his “sieve” — his procedure for deciding what topics are worth studying further — and why this sieve has so far filtered out homotopy type theory.
This made me think — and not for the first time — about why different communities with different attitudes toward “foundations” have trouble understanding each other. They argue, but the arguments aren’t productive, because they talk past each other.