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September 23, 2020

New Normed Division Algebra Found!

Posted by John Baez

Hurwitz’s theorem says that there are only 4 normed division algebras over the real numbers, up to isomorphism: the real numbers, the complex numbers, the quaternions, and the octonions. The proof was published in 1923. It’s a famous result, and several other proofs are known. I’ve spent a lot of time studying them.

Thus you can imagine my surprise today when I learned Hurwitz’s theorem was false!

Abstract. We present an eight-dimensional even sub-algebra of the 2 4=162^4=16-dimensional associative Clifford algebra Cl 4,0\mathrm{Cl}_{4,0} and show that its eight-dimensional elements denoted as X\mathbf{X} and Y\mathbf{Y} respect the norm relation XY=XY\| \mathbf{X} \mathbf{Y}\| = \| \mathbf{X} \| \| \mathbf{Y} \|, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Even more wonderful is that the author has discovered that the unit vectors in his normed division algebra form a 7-sphere that is not homeomorphic to the standard 7-sphere. Exotic 7-spheres are a dime a dozen, but those merely fail to be diffeomorphic to the standard 7-sphere.

Posted at 5:54 PM UTC | Permalink | Followups (23)

September 17, 2020

Special Numbers in Category Theory

Posted by John Baez

There are a few theorems in abstract category theory in which specific numbers play an important role. For example:

Theorem. Let S\mathsf{S} be the free symmetric monoidal category on an object xx. Regard S\mathsf{S} as a mere category. Then there exists an equivalence F:SSF \colon \mathsf{S} \to \mathsf{S} such that:

  • FF is not naturally isomorphic to the identity,
  • FF acts as the identity on all objects,
  • FF acts as the identity on all endomorphisms f:x nx nf \colon x^{\otimes n} \to x^{\otimes n} except when n=6n = 6.

This theorem would become false if we replaced 66 by any other number.

Posted at 2:07 AM UTC | Permalink | Followups (7)

September 16, 2020

Open Systems: A Double Categorical Perspective (Part 2)

Posted by John Baez

Back to Kenny Courser’s thesis:

One thing Kenny does here is explain the flaws in a well-known framework for studying open systems: decorated cospans. Decorated cospans were developed by my student Brendan Fong. Since I was Brendan’s advisor at the time, a hefty helping of blame for not noticing the problems belongs to me! But luckily, Kenny doesn’t just point out the problems: he shows how to fix them. As a result, everything we’ve done with decorated cospans can be saved.

Posted at 5:14 PM UTC | Permalink | Post a Comment

September 15, 2020

Symmetric Pseudomonoids

Posted by John Baez

The category of cocommutative comonoid objects in a symmetric monoidal category is cartesian, with their tensor product serving as their product. This result seems to date back to here:

Dually, the category of commutative monoid objects in a symmetric monoidal category is cocartesian. This was proved in Fox’s suspiciously similar paper in Cocomm. Coalg.

Posted at 8:41 PM UTC | Permalink | Followups (11)

September 7, 2020

Riccati Equations and the Projective Line

Posted by John Baez

Riccati equations are a natural next step after you’ve studied linear differential equations. Linear first-order ordinary differential equations look like this:

dydt=f 1(t)y+f 0(t) \frac{d y}{d t} = f_1(t) y + f_0(t)

Riccati equations look like this:

dydt=f 2(t)y 2+f 1(t)y+f 0(t) \frac{d y}{d t} = f_2(t) y^2 + f_1(t) y + f_0(t)

I think I finally get why they’re interesting. Riccati equations are to the projective geometry as linear first-order ordinary differential equations are to affine geometry!

Posted at 5:09 PM UTC | Permalink | Followups (14)

September 6, 2020

Three Phases of Continued Fraction Theory

Posted by John Baez

I don’t know much about continued fractions yet, so it’s too early to be describing historical phases of work on the subject, but I can’t resist doing it. I’ll talk about three:

  • The Greeks
  • Euler
  • Gauss

I won’t talk about general theories of continued fractions, like their connection to Pell’s equation, Calkin–Wilf trees and rational tangles, or the line of work from Gauss to Khinchin and beyond on the statistical properties of the continued fractions of ‘typical’ numbers, or the work of Pavlovic and Pratt on a characterization of [0,)[0,\infty) as the terminal coalgebra of some endofunctor on the category of totally ordered sets, which uses continued fractions. Indeed, I will not even mention these things, fascinating though they are. Instead, I’ll only talk about continued fractions that can be evaluated to give famous numbers or functions.

Posted at 12:26 AM UTC | Permalink | Followups (6)

September 3, 2020

Announcing the Johns Hopkins (Virtual) Category Theory Seminar

Posted by Emily Riehl

The Johns Hopkins category theory seminar is a “topics course” in intermediate/advanced category theory, aimed at mathematics graduate students who have taken a first course or who have read an introductory book. Talks are given by the local participants, often on material they are learning for the first time. Interruptions to ask questions are highly encouraged.

This semester, the category theory seminar will meet virtually, and my students have suggested I invite other category-learners to join them. The seminar will meet on Wednesdays from 21-23pm UTC (which is Thursday morning in certain parts of the world). The talk itself will take place from 21:15-22:15, with the rest of the time reserved for discussion and socializing.

The planned talks have been announced on where you can click on a link that conveys you directly into the zoom room. Even better: if you create a profile, the schedule will be displayed in your local time zone.

Out of a desire to organize a slimmed-down seminar - virtual talks being considerably more exhausting than IRL ones - all of the speakers in the seminar are Johns Hopkins affiliates. That said, we’d enjoy having the opportunity to get to know other budding category theorists elsewhere, so if you’re interested, please join us (with the warning that we may ask you to introduce yourself).

Posted at 8:10 PM UTC | Permalink | Followups (4)

Chasing the Tail of the Gaussian (Part 2)

Posted by John Baez

Last time we began working on a puzzle by Ramanujan. This time we’ll solve it — with some help from a paper Jacobi wrote in Latin, and also from my friend Leo Stein on Twitter!

Posted at 6:57 PM UTC | Permalink | Followups (11)

Making Life Hard For First-Order Logic

Posted by David Corfield

I mentioned in a previous post Sundholm’s rendition in dependent type theory of the Donkey sentence:

Every farmer who owns a donkey beats it.

For those who find this unnatural, I offered

Anyone who owns a gun should register it.

The idea then is that sentences of the form

Every AA that RRs a BB, SSs it,

are rendered in type theory as

z: x:A y:BR(x,y)S(p(z),p(q(z)), \prod_{z: \sum_{x: A} \sum_{y: B} R(x, y)} S(p(z), p(q(z)), where pp and qq are the projections to first and second components. We see ‘it’ corresponds to p(q(z))p(q(z)).

This got me wondering if we could make life even harder for the advocate of first-order logic – let’s give them the typed version to be generous – by constructing a natural language sentence which it would be even more awkward to formalise.

Posted at 1:52 PM UTC | Permalink | Followups (15)