This is a bit of a wild stab in the dark, and I know the readership of the forums may be teetering on limited, but I've wanted for as long as I can remember (at least a couple of months...), to understand a particularly complicated piece of maths. It's called seiberg-witten theory and it's ace.

It is an example of a particularly spiffy kind of maths called a gauge theory; originally developed for use in physics (specifically quantum field theory), gauge theory uses some rather trendy differential geometric machinery to model the local symmetries of a differentiable manifold, telling us something in the process of its global structure. In the 1980s Donaldson used a similar theory (yang-mills, of millenium prize fame) to prove some extraordinary results about differential topology. In particular, he was able to prove that admits infinitely many differentiable structures- that is, there are plenty of objects the shape of - but only one of them may be differentiated as we know it.

Seiberg witten theory has since been shown to prove a superset of Donaldson's results, and with greater ease. It represents for me the gateway to the exciting and unsolved world of guage theory, and having found (via the wikipedia article )some free and rather comprehensive notes on the subject, I couldn't help but feel the gate had been left open.

But reading on my own would be hard. If anyone is at all interested, and has a reasonable grasp of manifold theory/ differential geometry/ can understand at least 4 words in the wikipedia article, I would love to hear from you.

The dream is: set up a reading group, meet once before the holidays, once a weekish term 3- and get this bad boy nailed.

Notes are here btw.

You crazy little man. We talked about this, we really need to do some actual work next term. I strongly suggest waiting until the 3rd/4th year exams are over before you even start.

That sounds pretty good, I could try even though I don't think I have the necessary background. I had heard about this kind of stuff (though more from a physics point of view, plus what you told me a few days ago) and would love to learn stuff about it, if I'm able to.

P.S. Gauge Theory :P

P.S. Gauge Theory :P

Posted at 4am.
:P
Changed it now...

Slightly related, I wanted to ask for an explanation about something I read on Wikipedia

A fundamental fact of differential topology is that the notion of isomorphism in Top, PL, and Diff is the same in dimension 3 and below; in dimension 4 PL and Diff agree, but Top differs. In dimension above 6 they all differ.

Is it so fundamental that I could understand it ?

I believe that the affirmative answers to questions of agreement in these categories are all to do with simplical approximation theorems, theorems that allow maps and manifolds to be split up into tetrahedral regions without losing structure. They could certainly be said to be fundamental in their repercussions, and indeed any self-respecting differential topologist would be amiss if he or she were not more than familiar with their use; but that is not to say they're easy- in 2 dimensions simplical approximation is equivalent to the jordan curve theorem (simple closed cuves split the plane into an interior and exterior- another 'fundamental' that's a bugger to prove), and I have no idea about proofs in higher dimensions.

Higher dimensional failings are all, somewhat predictably, reliant on counterexamples- none of which came to light 'til the sixties.

So yeah, the results may be fundamental, but perhaps not in the way you'd expect...