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:
- a*(b*c) = (a*b)*c for any a,b,c in G (associativity)
- There is an element e such that for any a in G e*a=a*e=a (identity element)
- 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 Poincaré spent some time thinking about geometry, and formulating his ideas in a way that lay the ground stones to the modern study of topology. This was to a large extent done in his Analysis Situs. In his work he considers closed surfaces – i.e. surfaces without a border, and that are limited in their girth. To be specific, they need to be compact – which rules out things like the plane, but allows spheres, tori, Klein bottles and such entities. Poincaré considered the question of what essentially different surfaces are possible. We’re not interested hereby to distinguish between surfaces just because they happen to be scaled differently, but we are interest in distinguishing if they have differences that are independent of scaling factors.
To attack the question, Poincaré studied closed curves on the surface. A closed curve is something that is parametrized by the interval [0,1] and such that it starts and ends in the same point. So, mathematically, it’s a function , with f(0)=f(1).
The normal way to study these entities is to also require all the curves we’re looking at to start at one specific point x0, so we require f(0)=f(1)=x0. Should we have two such functions, f and g, we can compose them by taking h(t)=f(2t) during [0,0.5] and h(t)=g(2t-1) during [0.5,1].
Now, the structure we get this way ALMOST ends up being a group as we stated up at the beginning. It isn’t quite, since we actually see a difference between staying in one place for half the time and then rushing through the race course in double speed for the rest of the time; or for that matter, composing three functions two different ways ends up being different as well; we don’t think it is the same partition when we do
as when we do
However, both this associativity as well as the identity element (racing the course in double speed) and inverses (going first one way, then the other) can all be solved if we view two curves as equal if they can be continuously deformed into each other. So we basically end up with a deformation from f to g being a map H(x,t) parametrized, so that H(0,t)=f(t), H(1,t)=g(t), and everything is continuous in all directions.
In classical algebraic topology, we more or less stop at this point, say that two curves are homotopic if they have a homotopy (such an H) between them, and form a group of equivalence classes of curves, called the fundamental group. That’s all nice, but not quite what I want to go and do.
In fact, I’m very much interested in this homotopy of associativity. We have the two associativity versions (fg)h and f(gh), depicted above with the tables. And some homotopy H between them. However, when we then look at ways to associate four curves, we get chains of homotopies that need not be the same. To be specific, we can use our homotopies guaranteed from the first step to ensure that
((fg)h)i — (f(gh))i — f((gh)i) — f(g(hi))
((fg)h)i) — ((fg)(hi)) — f(g(hi))
where all the — are 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. If that’s the case, then this homotopy will give rise to the faces of a 3-dimensional polygon, where, again, the surface of the polygon represents homotopical paths that need not be equal, but may well be homotopic. This continues on, and on, and on, and a topological space where these homotopies are present all the way up is called an A∞-space.
With inspiration from the A∞-spaces, and because it pops up as the cohomology of such spaces (another, much nicer to calculate invariant than the fundamental group and its cousins, the homotopy groups – since homology and cohomology ends up being mostly linear algebra) there is a theory formulated by Jim Stasheff originally, and revitalized by among others Bernhard Keller dealing with algebras (i.e. algebraic structures where +, -, * and occasionally even / works) that have a set of higher multiplications that take any number of arguments, and not only 2 as normal multiplication does, and then returns something else from the algebra; and that obey axioms that are taken from these homotopies of homotopy paths we had in the last section. There is one axiom built on f(gh) — (fg)h, there is one axiom built on the homotopy between the two different paths between ((fg)h)i and f(g(hi)), and so on.
The really funky bit pops up when we combine this with cohomology. My advisor and I are interested in a way to do topology on groups; if we have a finite group – a finite set with an operation that fulfills the axioms from the beginning; we can build an algebra from it, that in some way represents the topological cohomology of a space that was built from the group in a welldefined manner. This cohomology carries some information about the original group, but loses some, so there may be infinitely many groups that give the same cohomology algebra.
However – if we also know the A∞ structure on the cohomology, if we know all these higher multiplications in addition to knowing the actual cohomology algebra, then we can reconstruct the original cohomology algebra from this data. This statement is at the core of the work done in the study of A∞-algebras, and 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.