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.

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 (17)

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 (26)

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 (3)

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 (22)

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)