Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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 (4)

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.

Posted at 2:05 PM UTC | Permalink | Post a Comment

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 (77)

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 (12)

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)

March 21, 2016

Prime Numbers and the Riemann Hypothesis

Posted by John Baez

I hope this great book stays open-access, but I urge everyone to download a free copy now:

It’s the best elementary introduction to the connection between prime numbers and zeros of the Riemann zeta function. Fun, fun, fun!

Posted at 5:12 AM UTC | Permalink | Followups (1)

Coalgebraic Geometry

Posted by Qiaochu Yuan

Hi everyone! As some of you may remember, some time ago I was invited to post on the Café, but regrettably I never got around to doing so until now. Mainly I thought that the posts I wanted to write would be old hat to Café veterans, and also I wasn’t used to the interface.

Posted at 5:00 AM UTC | Permalink | Followups (7)

March 19, 2016

The Most Common Prime Gaps

Posted by John Baez

Twin primes are much beloved. But a computer search has shown that among numbers less than a trillion, most common distance between successive primes is 6. It seems this goes on for quite a while longer…

Posted at 6:10 AM UTC | Permalink | Followups (12)

March 15, 2016

Weirdness in the Primes

Posted by John Baez

What percent of primes end in a 7? I mean when you write them out in base ten.

Well, if you look at the first hundred million primes, the answer is 25.000401%. That’s very close to 1/4. And that makes sense, because there are just 4 digits that a prime can end in, unless it’s really small: 1, 3, 7 and 9.

So, you might think the endings of prime numbers are random, or very close to it. But 3 days ago two mathematicians shocked the world with a paper that asked some other questions, like this:

If you have a prime that ends in a 7, what’s the probability that the next prime ends in a 7?

I would have expected the answer to be close to 25%. But these mathematicians, Robert Oliver and Kannan Soundarajan, actually looked. And they found that among the first hundred million primes, the answer is just 17.757%.

So if a prime ends in a 7, it seems to somehow tell the next prime “I rather you wouldn’t end in a 7. I just did that.”

Posted at 2:52 AM UTC | Permalink | Followups (18)

March 10, 2016

Category Theory Course Notes

Posted by John Baez

Here are the notes from a basic course on category theory:

Unlike my Fall 2015 seminar, this quarter I tried to give a systematic introduction to the subject. However, many proofs (and additional theorems) were offloaded to another more informal seminar, for which notes are not available. So, many proofs here are left as ‘exercises for the reader’.

Posted at 9:25 PM UTC | Permalink | Followups (4)

March 9, 2016

Category Theory Seminar Notes

Posted by John Baez

Here are some students’ notes from my Fall 2015 seminar on category theory. The goal was not to introduce technical concepts from category theory—I started that in the next quarter. Rather, I tried to explain how category theory unifies mathematics and makes it easier to learn. We began with a study of duality, and then got into a bit of Galois theory and Klein geometry:

Posted at 6:17 PM UTC | Permalink | Followups (5)

March 4, 2016

Hyperbolic Kac–Moody Groups

Posted by John Baez

Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras is connected to differential geometry and physics by the realization that these are the Lie algebras of central extensions of loop groups:

  • Andrew Pressley and Graeme Segal, Loop Groups, Oxford U. Press, Oxford, 1988.

  • Graeme Segal, Loop groups.

Indeed it’s not much of an exaggeration to say that central extensions of loop groups are to strings as simple Lie groups are to particles!

What comes next?

Posted at 6:47 PM UTC | Permalink | Followups (4)

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)