admissions of difficulty.

these past few days i've been "reading" a paper of Sullivan's from proceedings in a geometric topology conference held in georgia, sometime before 1980 (i think). i've been having three recurring thoughts about it.

1. in less than a dozen pages, he proves a result that took a one-semester course to present .. and the current plan is that i extend the result.
   
   yikes ...
   
   i wonder if i can actually do what i said i would do .. or thinking more positively, i wonder how far i'll get. [1]
   
   
2. this result is hard .. or at least, it is very hard for me.
   
   over these few years, i think i've developed a notion of what i am capable (or incapable) of understanding a particular notion at a given time and place. moreover, i might even detect why i might possibly understand something.
   
   for example, following seminar talks at gft [2] isn't too bad, if only because i sat through so many of them by now. compare this with when i was a first year: i can say with honesty that i was lost most of the time.
   
   i can't say that i understand sullivan's proof .. yet. it's a dangerous and worrisome thing when i ask myself, "why did he include this part, and why is this necessary?" because it often means that i really don't know what's going on: strategy, details, or otherwise.
   
   for example, i think i'm missing something that should even be obvious: i'm still not entirely sure where the "Lipschitz" comes from in this Lipschitz Structure Theorem. admittedly, it still seems somewhat magical.
   
   so .. there is a LOT of work to do .. and the scary thing is, by now i'm supposed to have worked on this project for over a year.
   
   
3. this result is pretty cool. i can't really explain why, but it feels like it was done right.
   
   i only have these impressions when i read certain authors and works, such as elias stein, lars ahlfors, john milnor, fred gehring, and others. conversely, i wish i could add m. gromov to that same list, but that would be a lie; i simply cannot understand him most of the time, and can't appreciate it.
   

[1] the theorem is: every topological n-manifold, n ≠ 4, admits a Lipschitz structure; that is, transition maps between charts can be improved to bi-Lipschitz homeomorphisms. moreover, the structure is unique up to such homeomorphisms of the manifold.

[2] geometric function theory (seminar), otherwise known as the wednesday seminar for the analytically-inclined @ um.