Michi’s blog » archive for 'Combinatorics'

High school topology restarting

  • November 16th, 2007

Today, I told my two bright students about abstract and geometric simplicial complexes, about the boundary map and the chain complex over a ring R associated with a simplicial complex Δ, and assigned them reading out of Hatcher’s Algebraic Topology.

The next couple of weeks will be spent doing homology of simplicial complexes, singular homology, equivalence of the two, neat things you can do with them; and then we’ll start moving towards a Borsuk-Ulam-y topological combinatorics direction.

I might end up pulling combinatorics papers from my old “gang” in Stockholm on graph complexes, and graph property complexes, and poke around those with them.

(104 words)

Enumerating the Saneblidze-Umble diagonal in Haskell

  • June 11th, 2007

IMPORTANT: Note that the implementation herein is severely flawed. Do not use this.

One subject I spent a lot of time thinking about this spring was taking tensor products of A∞-algebras. This turns out to actually already being solved - having a very combinatorial and pretty neat solution.

Recall that we can describe ways to associate operations and homotopy of associators by a sequence of polyhedra Kn, n=2,3,.., called the associahedra. An A∞-algebra can be defined as being a map from the cellular chains on the Associahedra to n-ary endomorphisms of a graded vector space.

If this was incomprehensible to you, no matter for this post. The essence is that by figuring out how to deal with these polyhedra, we can figure out how to deal with A∞-algebras.

This is a preview of Enumerating the Saneblidze-Umble diagonal in Haskell. Read the full post (2258 words)

looksay - today’s Haskell snippet

  • April 18th, 2007

nextLookSay = foldr (\xs -> ([length xs, head xs]++)) [] . group
lookSay = iterate nextLookSay [1]
 

Conway’s Look-and-say sequence

(23 words)

Borsuk-Ulam and West Wing

  • February 7th, 2006

In West Wing 4×20, CJ states that there are two antipodal points with identical temperature on the earth, as an argument why it should be possible to imagine that an egg could stand on its end at the spring equinox. This particular plotline also has her most emphatically claiming that this should not be possible at the autumn equinox. Why this particular physics is complete idiocy will be left as an exercise to the interested reader, and instead I will focus on the other claim.

This is, in fact, true. It’s a corollary to one of the prettiest theorem conglomerates I have ever seen: the Borsuk-Ulam theorem(s). Alas, I haven’t got my sources on it here at the moment, so I won’t give you the deep indepth survey I want to give; but I do want to give a bit of overview as to why the claim CJ supports her insane theory with is actually true.

This is a preview of Borsuk-Ulam and West Wing. Read the full post (609 words, 15 images)

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Michi is a recent PhD working in homological algebra. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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