Topological Methods in Engineering, Networks, and Data Analysis

048979: Selected Topics in Probability and Stochastic Processes: Winter, 2016/17.

  • Instructor: Robert J. Adler
  • Time: Tuesdays, 14:30-16:30.pic1
  • Place: Meyer (EE) Building, Room 353
  • Content: This course is intended to be as much motivational as educational, and is intended to introduce engineers and data analysts to the powerful tools of topology, one of the oldest and yet most active areas of mathematics. It will also serve to educate interested mathematicians in the usefulness of their subject.


As technology progresses systems become larger and larger and are built from more and more components. Similarly, and not for unrelated reasons, data bases of all kinds have exploded. Predicting the behaviour of the complete system, or understanding the overall data structure, is often hard to do based simply on knowledge of the components. For over a hundred years topologists have been working of the similar problem of going from local combinatorial data to global information, and have been remarkably successful. In recent years these two areas have begun to interact and it is the aim of this course to describe the basic theory along with some of the applications of this interaction.

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 which is expected to include both some theory and an application.




Once the semester starts, and I see how many (if any) students there are, and we decide on precisely how/when the grades/projects/presentations will occur, I will post a list of projects/papers/etceteras here.


The following outline is not cast in stone: I will almost definitely 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.

Course outline by week
1 What is topology? What can it offer to engineering and data analysis?
3-4 Combinatorial topology, simplicial complexes. Homology and the algebraic description of shape. Persistent homology and bar codes.
5-6 Applications to `Big Data’: Dimension reduction and manifold learning from a topological viewpoint. Efficiency and sample size issues.
7 Manifolds and differential topology. Morse theory.
8 Configuration spaces for robot arms, topology of random linkages, navigational complexity of configuration spaces.
9 Euler characteristic. Euler integration, and network coverage, sensor networks.
10 Random functions. Level crossings. fMRI and statistical image analysis. Euler characteristics and random functions.
11 Random geometric complexes, topological phase transitions.
12-13 Student presentations (which may also overlap with some of the earlier weeks and topics).