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August 30, 2009

This Week’s Finds in Mathematical Physics (Week 278)

Posted by John Baez

In week278 of This Week’s Finds, hear how astronomers in a high desert in Chile use an "artifical star" to take photos of incredibly high resolution, and how this helped them spot what’s happening on Betelgeuse. Read how red supergiants like Betelgeuse and Ras Algethi spew out huge amounts of dust, which eventually forms planets like ours. Watch a Moon-sized object crash into a Mercury-sized one in a "hypervelocity collision" in a distant solar system:

Learn the new way to make graphene, and read my history of the Earth - for physicists. And when you’re ready: dive into groupoidification!

Heck, this deserves to be big:


Posted at August 30, 2009 3:34 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 278)

Hi! My friends in the astronomy department here would be upset if I didn’t comment on their related work. Plus, there is another nice picture of worlds colliding at their press release: Astronomers discover the dusty remains of two terrestrial planets. Here is a reprint of the paper .

Be sure to check out the artist who constructs a lot of these images, Lynette Cook.

Posted by: stefan on September 2, 2009 6:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

I’m glad I mentioned your blog — otherwise this might have been the first This Week’s Finds that nobody bothered to comment on!

Anyway: while you didn’t mention it, this paper you pointed us to:

is about another planetary system where it seems a massive collision created a lot of hot dust — this time orbiting a binary star!

Ouch!

Posted by: John Baez on September 3, 2009 6:02 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

John wrote:

I’m glad I mentioned your blog…

Whoops! Wrong Stefan! You’re Stefan Forcey! I thought you were Stefan Scherer, the guy who blogged about colliding planets orbiting HD 172555.

So, you know astronomers who study this stuff. Cool!

Posted by: John Baez on September 3, 2009 6:52 PM | Permalink | Reply to this

collisions and groupoids

It is fun to hang out with these guys. Our department actually intersects with their research center, and so we shared a table at the academic fair recently. A giant poster of the above picture got lots of worried looks, before we pointed out that there are two sun-like stars.

The groupoidification paper is also very nice. I’m really hoping that as I read it there might occur to me some way of using the ideas to clarify some of the problems in combinatorial algebra!

As a shameless-but-relevant promotion, here are the latest preprints from our group:
Geometric combinatorial algebras: cyclohedron and simplex
shows how to make interesting algebras out of the vertices and faces of some familiar polytopes.
New Hopf Structures on Binary Trees (Extended Abstract)
demonstrates a bunch of algebraic structure on the vertices of a less familiar polytope sequence, the multiplihedra. The 2d and 3d versions of this shape are the commuting diagrams for morphisms of bicategories and tricategories. That fact isn’t mentioned in the above preprints, but all the categorical connections are discussed in one of last year’s papers:
Quotients of the multiplihedron as categorified associahedra .

Posted by: stefan on September 4, 2009 4:49 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

Hi Stefan,

I just opened your papers and the first thing that pops out are those fantastic illustrations. That alone is enough to entice me to have a closer look.

Differential graded algebras of geometric shapes is close to my heart. I come at it with the motivation of developing novel approaches to scientific computation. The shapes then represent primary building blocks for spaces and geometries that can describe systems ranging from black holes to a cell phone radiating next to a computer model of a human head.

Urs and I wrote a paper together a while ago that can be described as a (combinatorial) differential graded algebra of “diamonds”. A “2-diamond complex” is essentially a binary tree so I’ll definitely have to check out the second link you provided as well.

In the meantime, if you’re interested, you can see some of my thoughts, which are still in rudimentary form, here and maybe here.

Cheers!

PS: It’s Friday!

Posted by: Eric Forgy on September 4, 2009 5:37 PM | Permalink | Reply to this

Re: algebra and polytopes

Thanks! I’ve spent some time with your paper about causal graphs. There must be some connections here, ready to be exploited. The polytope sequences we study all share the feature of important lattice structures on their 1-skeleta (vertex-and-edge graphs). For instance, the associahedra realize the Tamari lattice:
.
This projects to the skeleta of the hypercubes, which realize as a lattice the subsets of {1,…,n}, ordered by inclusion. The map from associahedra to hypercubes is an algebra homomorphism. The graded algebra on the hypercube vertices is known as the algebra of quasi-symmetric functions, QSym.

Question: does your paper with Urs actually define a graded algebra on hypercube faces?
-Stefan
Friday it is, and I may be leaving the office soon, but will check back on Monday!

Posted by: stefan on September 4, 2009 7:26 PM | Permalink | Reply to this

Re: algebra and polytopes

Check out works of Devados and of Forcey involving and beautifully illustrating other polytopal projections
cf. starting from the permutahedron


whoops! I see this stefan is indeed Forcey
I’m sure his page will lead you to Devados as well

also how to realize various polytopes relevant to algebra by truncating e.g. hypercubes

Posted by: jim stasheff on September 7, 2009 2:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

Awesome.

Sometimes when I look at these geometric shapes I wonder if they are projections along some axis of a hypercube (they probably are). That would be neat because we could then construct the calculus on that hypercube and study fields on those shapes as projections of fields on the hypercubes. Or something…

Question: does your paper with Urs actually define a graded algebra on hypercube faces?

Yep. It is an associative noncommutative differential graded algebra of n-cubes. That aspect of it was originally formulated (to the best of my knowledge) by Dimakis and Mueller Hoissen (see this, for example). Their work is mostly with generic directed graphs, but we focused on directed graphs associated with directed hypercubes (which we called “n-diamonds”).

It might be too low brow for you, but I spelled out some basics on a binary tree here. I hope the references to finance are not too distracting. The binary tree has a special place in financial modeling so it is a natural application of this stuff.

Welcome to the nCafe and hope you join the nLab as well (if you haven’t). It’d be fun to try to work out some stuff together.

Posted by: Eric Forgy on September 4, 2009 8:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

I wonder if they are projections along some axis of a hypercube

You mean, whether an associahedron is a zonotope? (A zonotope is an affine projection of a hypercube, or equivalently a Minkowski sum of intervals.) This is not the case already for the associahedron K 5K_5, since it has pentagonal faces.

Posted by: Tobias Fritz on September 5, 2009 1:41 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

I hope you don’t mind corrections about English usage.

This is not the case already for …

“Already” strongly implies a change over time that is now complete. While you could consider going between the various associahedrons in order, that process is far too implicit to really count. I’d write this as “We can see this is in not the case even for the small associahedron …”

Posted by: Aaron Denney on September 9, 2009 11:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

I don’t see anything wrong with already. It makes perfect sense to go through the associahedra in order, since the further you go, the more complicated they get and the less likely they are to have nice properties. I’m pretty sure I’ve heard native English speakers using this idiom.

Posted by: Tim Silverman on September 10, 2009 11:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

Whatever, I even appreciate small tidbits of corrections in order to master the intricacies of the english language a little better.

Concerning the associahedra and their order, already the two-dimensional associahedron (a pentagon) is not a zonotope. The reason is that it is not centrally symmetric, while all the faces of a zonotope, including itself, are necessarily centrally symmetric.

Posted by: Tobias Fritz on September 10, 2009 12:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 278)

However, the associahedron is a Minkowski sum of simplices, as shown by Postnikov in his paper on generalized permutohedra .

Posted by: stefan on September 11, 2009 4:17 PM | Permalink | Reply to this

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