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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 (84)

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 (14)

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 (3)

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 (6)

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, the 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 (19)

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)