Last night I had an amazing time at Madison Math Circle, hanging out with a bunch of bright school kids, who were undaunted by either diseases or mathematics. Since I had quite a few questions after the lecture, I decided to post an overview so anyone interested can go over it themselves. For those of you who attended and have further questions, or even wish to explore mathematical dynamics on your own, feel free to leave a comment below or e-mail me at firstname.lastname@example.org and I’ll try to answer all your questions and provide some guidance for future.
I am currently attending 10th AIMS Conference on dynamics in Madrid, where I participated in the session on transport barriers in unsteady fluid flows. One of the talks in the session suggested that the ergodic quotient techniques for detecting coherent sets would not apply to finite-time setting, i.e., when ergodic averages are replaced by finite-time averages. This is, of course, of interest in cases when ergodic averages do not exist, so finite-time average is a potential resolution of this problem. I decided to investigate the potentially-problematic example given (2d linear saddle) using the code I made available at [bitbucket.org]. I found that in that case, ergodic quotient algorithm still works, even away from the asymptotic averaging limit.
Yesterday, I have given a presentation at AIMS Conference in Madrid, Spain. Braid Dynamics of Non-Periodic Trajectories [pdf]. Click on “Read more” in this post for the abstract.
I’ve given a short lecture on spectral graph theory in the Physical Applied Math group meeting. My notes are available as PDF.
The annotated reading list is below. I have copies of all the books listed so come and talk to me if you’re interested.
- An example of a common use of graph Laplacians as a stand-in for combinatorial optimization:
Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8), 888–905.
- On structure of eigenvectors:
Biyikouglu, T., Leydold, J., & Stadler, P. F. (2007). Laplacian eigenvectors of graphs (Vol. 1915). Berlin: Springer.
- THE monograph on spectral graph theory, a lot of material, but very readable.
Chung, F. R. K. (1997). Spectral graph theory (Vol. 92). Published for the Conference Board of the Mathematical Sciences, Washington, DC. Sample chapters on Fan Chung’s page.
- The random walk picture:
Lovász, L. (1996). Random walks on graphs: a survey. In Combinatorics, Paul Erdős is eighty, Vol.\ 2 (Keszthely, 1993) (Vol. 2, pp. 353–397). Budapest: János Bolyai Math. Soc.
- A good cross-section through applications of spectral graph theory:
Mohar, B. (1997). Some applications of Laplace eigenvalues of graphs. In Graph symmetry (Montreal, PQ, 1996) (Vol. 497, pp. 225–275). Dordrecht: Kluwer Acad. Publ.
This page http://earth.nullschool.net/ , created by Cameron Baccario, hosts a beautiful visualization of atmospheric and oceanic currents in close-to-real time. Go to the “About” page to see the explanation of what to look for at each pressure levels, what types of projections you could use, and what other data, like temperature and precipitation, you can overlay to help your understanding of Earth’s processes.
For those of you who would like to read more about the basics of applied ergodic theory that I talked about in 703 today, there are a few accessible papers and books below the fold. You are always very welcome to send me an e-mail or drop by my office (VV517) if you want to talk more about any of the papers or the topic in general.
Please e-mail me if any of the links below the fold don’t work, or if you have trouble locating the pdfs for the papers.
Here’s a simple demonstration of Arnold’s Cat Map. V.I. Arnold, a cool cat himself, figured out, decades before the age of the internet, that putting an image of a kitteh in your research vastly increases its appeal.