As technology progresses systems become larger and larger and are built from more and more components. The behaviour of the complete system is often hard to predict 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.
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 | What is topology? What can it offer to electrical and network engineering? |
| 2 | Manifolds and differential topology. Dynamic control. |
| 3-4 | Combinatorial topology, complexes. Circuit layout, dimension reduction and data compression. |
| 5-6 | Homology and the algebraic description of shape, bar codes. Dimension reduction and manifold fitting. |
| 7-8 | Topological invariants, Euler characteristic. Euler integration and network coverage, sensor networks. |
| 9 | Random functions. Level crossings. fMRI and statistical image analysis. |
| 10 | Morse theory |
| 11 | Euler characteristics and random functions. |
| 12 | Integral geometry, stereology. Building images from cross-sections. |
| 13 | More on random functions and Euler integrals. |
| 14 | Error estimates in manifold fitting. |
If you want extra information, you can reach me in the office at 8295753, at home at 8251794 (but not Shabbatot or Hagei Yisrael), or, most reliably, at robert@ee.technion.ac.il.
If you are reading this in hard copy rather than on the web, go to the Teaching section of my homepage to get the hyperlinks.