Dynamical Systems Seminar

 

Spring 2019

 

Title: Billiards in Polygons and related dynamical systems
Speaker: Robbie Robinson & participants
Date and Time: Friday, March 1st, 9:00–10:15am(or so)
Place:  Phillips 736

Abstract: We will continue our discussion about the dynamics of billiards in rational polygons. We will discuss and compare to related dynamical systems: rotations, interval exchange transformations, substitutions and geodesic flows on polygons.


Title: Introduction to billiards

Speaker: Robbie Robinson, GWU Math 
Date and Time: Friday, February 22nd, 9:00–10:15am
Place:  Phillips 736

Abstract: We will begin our discussion about the topic of the dynamics of billiards: Think of a billiard (i.e, pool) table whose boundary is a piecewise smooth simple closed curve. A ball moves (forever) around the interior, observing the angle of incidence equals angle of reflection rule each time it encounters the edge. Does the motion ever repeat (called periodic motion) or not? Does it go everywhere, or is it confined to a few parts of the table? How much time does it spend in one part of the table compared to another? I will begin with a general survey of what is known about these questions. Then we will switch to the special case of polygonal tables with rational angle corners. This is currently a very active research area related to “Teichmuller” theory. For the rest of the year, we will try to learn a little about it.


Title: On Penrose tilings and tiling dynamical systems: VI

Speaker: Robbie Robinson, GWU Math 
Date and Time: Friday, February 15th, 9:00–10:15am
Place:  Phillips 736

Abstract: The set of Penrose tilings is “almost” parameterized by a 4-torus. The R^2 translation action is product of a pair of suspensions of Sturmian dynamical systems. The inflation map is (almost) a hyperbolic total automorphism, with the Markov partition drawing the Penrose tiles. 


Title: On Penrose tilings and tiling dynamical systems: III
Speaker: Robbie Robinson, GWU Math 
Date and Time: Friday, February 8th, 9:00–10:15am
Place:  Phillips 736

Abstract: This time I discuss de Bruijn’s theorem on the structure of Penrose tilings (as duals of Penrose “pentagrams”) and show how to fit this into a dynamical systems context. This is lecture III in a series of IVTitle: On Penrose tilings and tiling dynamical systems: II


Speaker: Robbie Robinson
Date and time: Friday, February 1, 9-10:15am
Place: Phillips736

Abstract: Continuation of series of talks in which I present the ideas about Penrose tilings due to Penrose, de Bruijn, and Conway, as well as some of my own work on Penrose dynamics. These lectures are based on a series of lectures I gave about 20 years ago at Tsuda College in Tokyo. But work on aperiodic tilings and quasicrystals continues to be active research area under the title of “Aperiodic Order”. Later in the semester, the topic will switch to conformal geometry of billiards As in the Fall, the seminar will feature talks by participants as well as several outside speakers. For more information, see https://blogs.gwu.edu/robinson/2019/01/18/dynamical-systems-seminar-spring-2019/


Title: On Penrose tilings and tiling dynamical systems: I

Speaker: Robbie Robinson
Date and time: Friday, January 25, 9-10:15am
Place: Phillips 736
 
Abstract: 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 pseudo-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 space. The mural across from the math office is a piece of Penrose tiling. 
 
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 that is 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's structure theorem to model Penrose dynamics as a total rotation action.
 
In this talk (and the next few weeks in the seminar) I will present the work of Penrose and de Bruijn, as well as my own work on Penrose dynamics. These talks are based on a series of lectures I gave about 20 years ago at Tsuda College in Tokyo. But work on aperiodic tilings, their dynamics and the relation to quasicrystals continues to be active research area under the title of “aperiodic order”.
 
Once we finish Penrose tilings in a few weeks, 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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