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Moduli spaces of surfaces: Billiards and dilation surfaces

Periodic billiard paths on regular polygons- Diana Davis (Swarthmore College)

Abstract: Mathematicians have understood periodic billiards on the square for hundreds of years, and my collaborator Samuel Lelièvre and I have understood them on the regular pentagon for about five years now. During the COVID-19 pandemic, I have been in France, working with Samuel to extend our understanding to all regular polygons with an odd number of sides. In this talk, I'll briefly explain results and techniques for the square and pentagon, and then show lots of nice pictures of billiards on polygons with more than 5 sides, that we have created recently.

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An invitation to dilation surfaces- Selim Ghazouani (University of Warwick)

Abstract: In this video talk, I will introduce dilation surfaces, their moduli spaces and related foliations on surfaces and then try to give some motivation for a range of open problems.

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You can “hear” the shape of a polygonal billiard table- Chandrika Sadanand (University of Illinois Urbana Champaign)

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge-itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

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7 replies on “Moduli spaces of surfaces: Billiards and dilation surfaces”

Hello! I am a postdoc at UIUC. I hope you are all enjoying the conference – I certainly am. I would love to hear your thoughts and questions after seeing my talk. Please ask and comment!

Hi Chandrika!

Enjoyed your talk! Can these results be promoted to the hyperbolic setting? More precisely, if I have a polygon in the hyperbolic plane, and billiard trajectories follow hyperbolic geodesics, do you expect these same results to hold?

Hi Sunrose, nice question. We have been looking into this and we are now writing it up. Initially, we were able to prove the rigidity result for hyperbolic cone surfaces with irrational cone angles, and then hypothesized that it must be true for the rational angle hyperbolic cone metrics as well. One difficulty in adapting our proofs to the hyperbolic case was that families of parallel lines in the hyperbolic plane behave very differently than in the Euclidean plane. As we figured it out, I was pleasantly surprised to find that the hyperbolic case has in some sense, a little less rigidity than the Euclidean case (I thought it should be more rigid, as hyperbolic things often are). The flexibility of the metric in the non-rigid cases is very interesting, it comes from a family of metrics on a quotient orbifold. Overall, the generic case is rigid, and you can hear the shape of a generic hyperbolic polygonal billiard table!

Hi all! I am really enjoying watching the talks. I am finishing up as a Visiting Assistant Professor at Swarthmore College, and also as an invited research visitor at the Institut des Hautes Études Scientifiques outside of Paris. I enjoyed talking with several of you at my office hour on Monday night, and I’d love to talk with any of you further!

Hi Diana,

I asked a similar question like this to Chandrika as well.

Do you know of any work that has been done in the hyperbolic analogue of this, i.e. billiards on a hyperbolic polygon where trajectories follow hyperbolic geodesics? Unfolding should still work in that case, right?

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