math class
Jan. 11th, 2010 11:30 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslI'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.
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.