Thursday, May 22, 2008

Postcoital tristesse

As I read more and more about this Delignes article I start thinking about the time when I will completely understand this beauty. Isnt it sad that after the end I will move on to some other problem, just taking the ideas from this relationship, to apply in some other cases. Yes she is beautiful. Not the most beautiful thing I met this year, but she is so beautiful that everytime I try to understand her, I face a massive challenge. So I have to change my position, change my approach, but not for a single moment she allows me to take my attention away from her. She plays silly tricks by suddenly closing all the windows and not letting me to understand her. Then there lots of cigarette gets lit, the grand worship of this beauty begins. And as usual after few hour of maska marowing, there is a ray of light, some different angle of looking at it. Now I have a general outline of how she thinks and how she looks. It took long time to make this outline. For the sake of mathophiles this is the outline :
She is like something we know already, or we think we know already, obviously we know what are the fundamental groups of projective line minus three points in algebraic and topological cases. But then she is not what she looks like. She is still out of our reach of understanding and I will be really happy if she remains like that forever. What Delign starts thinking that like the unification of cohomology theories ( motives) , can we see the different fundemental groups related to some motive. So for that he first unifies all the cohomology theory in a naive way, called system of realisation. Then in this category of system of realisation we can define fundamental group as this category is a Tanakian category. Then he gives description of the motivic fundamental group and then the motivic fundamental group associated with projective line minus three points turns out to be the iterated extensions of the Tate motive Q(n). If the ultimate aim of this study is not exciting enough then one should know the things that we get when we are following this path. There are algebraic connections, algebraic monodromy and foliation floating around, then for Tate motives you will get a beautiful Zeta function, many number theoritic properties gets identified with differential geometry. I haven't understood most of the things yet. But still going good. The opening of the paper gives me a lot of pleasure, where Delign writes that much of good things related to projective line minus three points are still undiscovered. That is very unwestern, where they always claim of comeplete knowledge of things , and on basis of those propagandas they classify objects as if this world and all the objects here are their baap ka jaigir. Hope to finish the Paper tonight, but feeling sad to leave it.

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