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Michi’s blog » A∞ for the layman

 A for the layman

  • November 7th, 2006
  • 4:18 pm

I recently had reason to describe my PhD work in complete laymans terms while writing letters to my grandparents. This being a good thing to do in order to digest your ideas properly, I thought I might try and write it up here as well.

It will, however, push through some 100-odd years of mathematical development rather swiftly. Try to keep up – I will keep it as light as I can while not losing what I want to say.

Algebra

In the 19th century, a number of different mathematical efforts ended up using more or less precisely the same structures, though not really recognizing that they were the same. This recognition came with Cayley, who first brought the first abstract definition of a group.

A group is a set G of elements, with a binary operation *, such that the following relations hold:

  1. a*(b*c) = (a*b)*c for any a,b,c in G (associativity)
  2. There is an element e such that for any a in G e*a=a*e=a (identity element)
  3. For any a in G, there is an element a’ in G such that a’*a=a*a’=e (inverses)

This turns out to be just what you needed to merge the studies of symmetries in geometric objects with the study of solvability of polynomial equations, and a number of other various ideas that were floating around. I will put this all on hold for a while, and then return to it at the end of this rant.

Topology

At the end of the 19th century and beginning of the 20th century, Henri Poincar

6 People had this to say...

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  • chuck
  • February 3rd, 2007
  • 17:40

I let this one languish in my RSS reader for a long time, meaning to finally sit down and read it so I could finally start to understand a little of what you’ve been on about. I finally read it this morning and really enjoyed it. Kind of makes me want to go back to school and study a little of this myself.

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BTW, you should be able to get é to render properly across browsers using & eacute ; (no spaces).

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To be specific, we can use our homotopies guaranteed from the first step … homotopies between the differently associated thingies. And these two different paths need in no way be equal. But, if things are nice enough, there’s going to be a homotopy between these paths.

Just to check: these paths are not the typical path γ: [0,1] → ℝ or whatever, they are sequences of homotopies. Right?

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Made you a picture to go with this blog post. http://25.media.tumblr.com/5f70242699154f91b4db8f039285e3ae/tumblr_mpch37V1MA1qc38e9o1_1280.jpg

I can email you the .odg if you want to mess with positioning. Feel free to post this anywhere on your blog.

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my current plans for my PhD is to see whether I can get all the things the theory promises to work for the specific case of group cohomology. There are theorems proven already that it should, probably, work; but this is a case that simply has not been considered so far. Thus just sitting down and working things through may be a significant advance of the state of knowledge in itself.

Very interesting. Now it’s 7 years later; how did that work out?

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