A∞ for the layman

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.


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.


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