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
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 (as well as all the other complex simple Lie algebras, except ) 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 , 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 .
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 . I’ll write when is an edge going from the vertex to the vertex . We get a vector space of 0-chains , which are formal linear combinations of vertices, and a vector space of 1-chains , which are formal linear combinations of edges. We also get a boundary operator
sending each edge to the difference . A 1-cycle is 1-chain with . 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:
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 such that 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…