Probability and Stochastic Processes for Persistent Homologists

Thursday, October 3, 2013 - 9:00am - 10:15am
Lind 305
Robert Adler (Technion-Israel Institute of Technology)
My first working assumption for this tutorial will be that the participants know something about persistent homology (PH). In particular, they know that PH is associated with filtrations, the two most common being the filtrations of upper or lower level sets of smooth functions and filtrations arising from building simplicial (Rips, Cech) complexes over point cloud data.

My second assumption is that the participants would also like to understand what happens in situations in which the underlying structure, whether it be a function or a point cloud, is random. (The motivation for such a desire should come from the realisation that data analysed by TDA is typically sampled, and so random.)

Based on this, I plan to give a quick run through a number of basic topics in Probability and Stochastic Processes, all of which are necessary (and maybe even sufficient) for understanding most issues of randomness for TDA. The topics that I plan to cover will include:

1: Limit theorems in Probability (laws of large numbers and the central limit theorem)
2: Random walk and Brownian motion (and why they are not terribly relevant to TDA)
3: Poisson and other point processes (without which there is no way to understand point clouds)
4: Gaussian random fields (which provide a powerful model for random Morse functions)

The tutorial will be blackboard based and so topic-flexible. Thus participants should bring their favourite questions with them, or, better still, send them to me ahead of time.