Can Probability and Topology Combine for an Exciting Research Topic?

[Paparazzi]

I need to write a paper on something to do with (general) topology, and we are encouraged to try and relate it to something that we enjoy. I really like probability (at least basic probability + stochastic processes), and I'm wondering if someone might suggest topic(s) that might be of interest relating the two fields.

Thanks a lot.

 

[quasar987]

There are various notions of convergence in probability. Some come from a topology, others not. Some type of convergence imply some others. This could make for a fun exposition.

http://en.wikipedia.org/wiki/Convergence_of_random_variables

 

[lavinia]

I need to write a paper on something to do with (general) topology, and we are encouraged to try and relate it to something that we enjoy. I really like probability (at least basic probability + stochastic processes), and I'm wondering if someone might suggest topic(s) that might be of interest relating the two fields.

Thanks a lot.

Solutions of the heat equation on manifolds is a profound subject. Heat flow is just Brownian motion.

 

[Bacle]

There is also an interesting chapter in Royden on Topology and Measure,
describing ways to assign a measure ( a sigma-algebra) on spaces with
only a topology defined on them.

 

[quasar987]

This is also done in many other places and it is called the Borel (sigma-) algebra.

http://en.wikipedia.org/wiki/Borel_algebra

 

[arsenal997]

A new topic in that area is the stochastic study of the fractals with probability and measure theory, in that sense you might do a topological study of the fractals

 

[Jamma]

I think that you picked some of the hardest two subjects of mathematics to actually combine, but good luck, anything you do come up with would be very nice. As others have already said, the most obvious connection I feel is via probability measures, although these are more regularly used for topological dynamics than just general topology itself.