Ergodic quotient in a finite-time, expanding context

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.
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Presentation: “Braid Dynamics of Non-Periodic Trajectories”

Yesterday, I have given a presentation at AIMS Conference in Madrid, Spain. Braid Dynamics of Non-Periodic Trajectories [pdf].

Basics of spectral graph theory

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.

  1. 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.
  2. On structure of eigenvectors:
    Biyikouglu, T., Leydold, J., & Stadler, P. F. (2007). Laplacian eigenvectors of graphs (Vol. 1915). Berlin: Springer.
  3. 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.
  4. 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.
  5. 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.
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Earth Wind/Ocean Map

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.

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MATH319: Techniques in ODEs

The website for the Spring 2014 course MATH319 at UW-Madison is online.

Basic readings in ergodic theory

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.

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Joining UW–Madison

Van Vleck Hall, 9th floor lounge

Van Vleck Hall, 9th floor lounge

In Aug 2013 I have started a Van Vleck VAP (postdoc) position at UW-Madison math department, working with Jean-Luc Thiffeault. My main project deals with braid theory of dynamical systems. For more details, please see the research page.