math class

[gusl]

I'm consider taking a math class, and it can't be Differential Geometry due to a conflict, so there's:

functional analysis, i.e. analysis in which each point is uncountably-infinite-dimensional (and can inherit properties that come from being a function)

* Banach spaces
* Operator spaces: strong, weak, and weak^* topologies
* Hilbert spaces and their geometry
* Operators on Hilbert spaces: self-adjoint, bounded, compact
* Hahn-Banach theorem, open mapping theorem, closed graph theorem
* Spectral theory for bounded operators
* Fredholm theory for bounded operators

Although I like the flavor of these topics, I don't know how they can be useful. I can see myself getting sucked in by mathematical curiosity from miles away... it's like the song of a siren promising to bring me back with more knowledge... but no, this Markov Chain is transient! I'm not a mathematician because I think there are more direct ways to do interesting research... (rule of thumb: if a piece of math depends on the Axiom of Choice for something more than proof-grease, it's not for me)

The lecture today was about the separability of disjoint convex sets. The pictures on the board made this look like just basic geometry, but since the definitions were wrt function spaces, the theorems were more general (i.e. about functionals, rather than functions). And did I mention this looks like a great field in which to use lambda calculus notation?


stochastic processes
* Random walks
* Markov chains (discrete and continuous time)
* The Poisson process
* Martingales
* Convergence of random variables
* Brownian motion and diffusions

It looks somewhat easier, as I'm already familiar with a lot (maybe most) of the material (mostly from just being a geek), but it doesn't hurt to go through it in more rigor. It is taught by a probabilist (who are, as I've learned, a very different tribe from statisticians!). Again I fear that it too may be too abstract to be worth the distraction.

Comments

[http://users.livejournal.com/cunctator_/]

How can functional analysis not be useful enough if differential geometry was even considered? DG seems much more abstract to me.

[gustavolacerda.livejournal.com]

Differential Geometry is useful for computing distances in manifolds!
I don't yet know what functional analysis is useful for.

[http://users.livejournal.com/cunctator_/]

Establishing convergence for approximation schemes? I think it`s useful in numerical analysis and even statistics at some point (empirical processes).

[gustavolacerda.livejournal.com]

Is empirical process theory concerned with anything more than the consistency of density estimators? Can you guess how functional analysis would be useful there?

[http://users.livejournal.com/cunctator_/]

Not only density, but all kinds of wicked estimators (notably in econometrics) and inference is a concern too - functional CLT appears to be the central object of study.
A whole course might be an overkill, but somewhere deep enough it bogs down to something like this: http://www.springer.com/math/probability/book/978-3-540-52013-9 so probably there is an intersection at some point.