Topological Properties of Random Systems

Instructor: Robert J. Adler

Time: Tuesdays, 09:30-11:30

Place: Meyer (EE) Building, Room 354.

Content: Over the past decade there has been a significant expansion of activity in applying the techniques and theory of algebraic topology to real world problems. The expression “applied algebraic topology” is no longer an oxymoron! This expansion has generated new mathematical theory, new computational techniques, and even commercial startups.

One aspect of topology having come out of its closet has been its interfacing with probability, stochastic processes, and statistics. This course is meant to introduce participants to work going on at these exciting interfaces.

The course will aim for breadth rather than depth (see the ambitious list of topics below) but each student will need to develop depth in at least one topic for an end of semester project/presentation.

Textbooks: The material will come from a variety of sources, most of them recent papers. However, some book sources are Robert Ghrist’s lecture notes Elementary Applied Topology and my book with Jonathan Taylor Random Fields and Geometry. A more user-friendly version of this is in my lecture notes Topological Complexity of Smooth Random Functions, a pre-publication version of which can be found here.

Grade: The final grade will (probably) depend on a combination of projects and presentations given by students at the end of the semester.

Prerequisites: It is to be assumed that few, if any, participants will have previous expertise in both stochastics and topology, and so the necesssary background will be developed as the course progresses. Nevertheless, undergraduate algebra and multivariate calculus, and a basic knowledge of stochastic processes and probability will be assumed, as well as either that undefinable quality known as “some mathematical maturity”, or the engineer’s knack (defined here) for putting pieces together to build a working system, even when he doesn’t know exactly how each piece functions.

Mainly, you will need patience to learn some new things and enthusiasm to piece a lot of quite different topics together.

Disclaimer: A knowledgeable topologist, probabilist, or statistican looking at the areas in which she specialises in the list of topics below will probably be aghast, claiming that there is enough meat in each of the five groupings for at least half a semester, if not a full one.

This is true, which is why there is the en passant disclaimer above that “the course will aim for breadth rather than depth”. I will feel the course has been successful if, at its completion, participants from probability and statistics background will enrol for a serious topology course, and the topologists will decide to learn some more completely presented probability and statistics.

More propaganda along these lines, more background motivation, and a description of why we need more linguists capable of speaking fluent Topologish, Stochastish, and Statistish, can be found on my web page


The following outline is not cast in stone: I will probably change it depending on who takes the course and what we turn out to be interested in, but it should give you an idea of to where we shall be heading.

Weeks Topic
1-2 I: Introduction to topology for discrete systems
Simplicial complexes: homology, Betti numbers, Euler characteristic.
Integral geometry: Steiner, Crofton, kinematic formulae.
Minkowski functionals.
Evaluations, Hadwiger’s theorem.
3-5 II: Discrete random systems
Erdos-Renyi random simplices: Homological phase transitions and limit theorems.
Cech complexes built over stationary point processes: Limit theory for Betti numbers.
6-7 III: Introduction to topology for continuous systems
Stratified manifolds.
Critical points and Morse theory. Betti numbers and the Morse inequalities.
Weyl’s tube formula, curvature integrals, Gauss-Bonnet theorem.
Lipshitz-Killing curvatures and Gaussian Minkowski functionals.
8-11 IV: Continuous random systems
Gaussian and related random fields
The generalised Kac-Rice formula for vector fields
The Gaussian kinematic formula
Gaussian extrema from a topological viewpoint. Slepian models.
Another look at discrete systems, via critical points and Morse theory.
12-13 V: Applications
Persistent homology and topological data analysis.
Euler integration, sensor networks and signal detection.
The Gaussian kinematic formula in brain imaging and cosmology.
Statistical estimation of topological invariants, manifold learning.