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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 Δ n\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:

Triangle with some curves

(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.

Posted at 6:30 PM UTC | Permalink | Followups (22)

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.

Posted at 4:12 AM UTC | Permalink | Followups (16)

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.

Posted at 3:39 AM UTC | Permalink | Followups (25)

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:

For an event which delves back also to pre-HoTT days, try my

Posted at 12:57 PM UTC | Permalink | Followups (2)

May 12, 2016

E8 as the Symmetries of a PDE

Posted by John Huerta

My friend Dennis The recently gave a new description of the Lie algebra of E 8\mathrm{E}_8 (as well as all the other complex simple Lie algebras, except 𝔰𝔩(2,)\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.

This is a story with deep roots: it goes back to two very different models for the Lie algebra of G 2\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 E 8\mathrm{E}_8.

Posted at 6:30 PM UTC | Permalink | Followups (18)

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.

Posted at 3:51 PM UTC | Permalink | Followups (16)

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”.

Posted at 7:30 PM UTC | Permalink | Followups (19)

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 Γ\Gamma. I’ll write e:vwe : v \to w \, when ee is an edge going from the vertex vv to the vertex ww. We get a vector space of 0-chains C 0(Γ,)C_0(\Gamma,\mathbb{R}), which are formal linear combinations of vertices, and a vector space of 1-chains C 1(Γ,)C_1(\Gamma,\mathbb{R}), which are formal linear combinations of edges. We also get a boundary operator

:C 1(Γ,)C 0(Γ,) \partial : C_1(\Gamma,\mathbb{Z}) \to C_0(\Gamma,\mathbb{Z})

sending each edge e:vwe: v \to w to the difference wvw - v. A 1-cycle is 1-chain cc with c=0\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.

Posted at 11:30 PM UTC | Permalink | Followups (19)

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++24 2=70 2 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 nn such that 1 2+2 2++n 21^2 + 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…

Posted at 6:52 PM UTC | Permalink | Followups (1)

April 30, 2016

Relative Endomorphisms

Posted by Qiaochu Yuan

Let (M,)(M, \otimes) be a monoidal category and let CC be a left module category over MM, with action map also denoted by \otimes. If mMm \in M is a monoid and cCc \in C is an object, then we can talk about an action of mm on cc: it’s just a map

α:mcc\alpha : m \otimes c \to c

satisfying the usual associativity and unit axioms. (The fact that all we need is an action of MM on CC to define an action of mm on cc 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 Set\text{Set} on objects in ordinary categories,
  • actions of monoids in Vect\text{Vect} (that is, algebras) on objects in Vect\text{Vect}-enriched categories,
  • actions of monads (letting M=End(C)M = \text{End}(C)), and
  • actions of operads (letting CC be a symmetric monoidal category and MM 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 MM on a category CC correspond to monoidal functors MEnd(C)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 mm under such a monoidal functor. In other words, monads on CC are the universal monoids which act on objects cCc \in C in the above sense.

Corresponding to this notion of action is a notion of endomorphism object. Say that the relative endomorphism object End M(c)\text{End}_M(c), if it exists, is the universal monoid in MM acting on cc: that is, it’s a monoid acting on cc, and the action of any other monoid on cc uniquely factors through it.

Posted at 1:40 AM UTC | Permalink | Followups (7)

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.

Posted at 4:54 PM UTC | Permalink | Followups (14)

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.

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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:

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.

Posted at 7:43 PM UTC | Permalink | Followups (79)

March 24, 2016

E8 Is the Best

Posted by John Baez

As you may have heard, Maryna Viazovska recently proved that if you center spheres at the points of the E 8\mathrm{E}_8 lattice, you get the densest packing of spheres in 8 dimensions:

• Maryna S. Viazovska, The sphere packing problem in dimension 8, 14 March 2016.

The E 8\mathrm{E}_8 lattice is

E 8={x 8(+12) 8: i=1 8x i2} \mathrm{E}_8 = \left\{x \in \mathbb{Z}^8 \cup (\mathbb{Z}+ \frac{1}{2})^8 \; : \;\, \sum_{i = 1}^8 x_i \in 2 \mathbb{Z} \right\}

and the density of the packing you get from it is

π 42 44!0.25367 \frac{\pi^4}{2^4 \cdot 4!} \approx 0.25367

Using ideas in her paper, Viazovska teamed up with some other experts and proved that the Leech lattice gives the densest packing of spheres in 24 dimensions:

• Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska, The sphere packing problem in dimension 24, 21 March 2016.

The densest packings of spheres are only known in dimensions 0, 1, 2, 3, and now 8 and 24. Good candidates are known in many other low dimensions: the problem is proving things, and in particular ruling out the huge unruly mob of non-lattice packings.

For example, in 3 dimensions there are uncountably many non-periodic packings of spheres that are just as dense as the densest lattice packing! There are also infinitely many periodic but non-lattice packings that are just as dense.

In 9 dimensions, the densest known packings form a continuous family! Only one comes from a lattice. The others are obtained by moving half the spheres relative to the other half. They’re called the ‘fluid diamond packings’.

In high dimensions, some believe the densest packings will be periodic but non-lattice.

For a friendly introduction to Viazovska’s discoveries, see:

• Gil Kalai, A breakthrough by Maryna Viazovska leading to the long awaited solutions for the densest packing problem in dimensions 8 and 24, Combinatorics and More, 23 March 2016.

I’m no expert on this stuff, but I’ll try to get into a tiny bit more detail of how the proofs work.

Posted at 4:58 PM UTC | Permalink | Followups (13)

March 23, 2016

The Involute of a Cubical Parabola

Posted by John Baez

In his remarkable book The Theory of Singularities and its Applications, Vladimir Arnol’d claims that the symmetry group of the icosahedron is secretly lurking in the problem of finding the shortest path from one point in the plane to another while avoiding some obstacles that have smooth boundaries.

Arnol’d nicely expresses the awe mathematicians feel when they discover a phenomenon like this:

Thus the propagation of waves, on a 2-manifold with boundary, is controlled by an icosahedron hidden at an inflection point at the boundary. This icosahedron is hidden, and it is difficult to find it even if its existence is known.

I would like to understand this!

I think the easiest way for me to make progress is to solve this problem posed by Arnol’d:

Puzzle. Prove that the generic involute of a cubical parabola has a cusp of order 5/2 on the straight line tangent to the parabola at the inflection point.

There’s a lot of jargon here! Let me try to demystify it. (I don’t have the energy now to say how the symmetry group of the icosahedron gets into the picture, but it’s connected to the ‘5’ in the cusp of order 5/2.)

Posted at 4:50 PM UTC | Permalink | Followups (21)