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Dynamical Systems Seminar, Spring 2019

Overview

Meets Fridays, 9:00am-10:15am (or sometimes until 11:00) in Phillips 736. Include talks by Prof. Robinson, talks by participants, and talks by visitors. The current seminar schedule will appear on the GW Math Seminar webpage:  https://math.columbian.gwu.edu/dynamical-systems-seminar

For the first three weeks I will talk about Penrose tilings and tiling dynamical systems: Penrose tilings were discovered around 1976 by Sir Roger Penrose as he was trying to see how close he could come to tiling the plane by regular pentagons. Penrose tilings are aperiodic tilings that are nevertheless in some sense almost periodic, and they inherit a pentagonal (quasi) symmetry from Penrose’s pentagons. Penrose tilings were popularized by Martin Gardner, who reported on some work on them by John H. Conway. The big advance in understanding these remarkable tilings, however, came with N. G. de Bruijn’s 1981 papers “Algebraic Theory of non-periodic tilings of the plane I & II” which showed how to interpret Penrose tilings as a 2-dimensional slice through  5-dimensional  Euclidean space. The mural across from the math office is a piece of Penrose tiling. A section is shown to the right. The blue-grey tiling at the top of the page is not a Penrose tiling, but rather different aperiodic tiling with 4-fold rotational (quasi) symmetry.

In the late 1980’s Penrose tilings were proposed by the U. Penn physicists Levine and Steinhardt as a model for a newly discovered state of matter called quasicrystals. Like quasicrystals, Penrose tilings can have a 5-fold rotational symmetry forbidden for ordinary crystals. I learned about Penrose tilings as a postdoc at Penn, and realized many of the ideas in the theory have a dynamical systems interpretation. My 1996 Transactions paper “The dynamical properties of Penrose tilings” showed how to use de Briujn structure theorem to model Penrose dynamics as a total rotation action.

In these talks, I will present the ideas of Penrose and de Bruijn, as well as my own work on Penrose dynamics. These talks are based on a series of lectures that I gave about 20 years ago at Tsuda College in Tokyo. But work on aperiodic tilings, dynamical systems, and quasicrystals continues to be active research area under the title of “aperiodic order”.

The entire series:

Once we finish Penrose tilings by mid February or so, or the seminar will revert to the subject of the year: Dynamics on surfaces (I will even tell you how to fit Penrose tilings into this context).  We will start working through the paper The conformal geometry of billiards, Laura de Marco, BAMS 48, 2011, http://www.ams.org/journals/bull/2011-48-01/S0273-0979-2010-01322-7/home.html

As before, the seminar will include talks by participants as well as several outside speakers.

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