Abstract:

A billiard trajectory in a polygon P in the Euclidean plane is the path of a particle inside P, following straight lines until it encounters a side, and then bouncing off so that the angle of reflection equals the angle of incidence.  A generic trajectory never encounters the corners in forward or backward time, and so produces a biinfinite symbolic coding: the itinerary of sides encountered  by the trajectory.  A natural question studied by Bobok and Troubetzkoy about 15 years ago is whether one can recover the shape of the polygon from the set of all itineraries: the symbolic coding.  They observed that two polygons with vertical and horizontal sides that differ by an affine transformation have the same codings, and proved that among rational polygons (those whose interior angles are rational multiple of pi), this essentially accounts for the only ambiguity.  I'll describe work with Duchin, Erlandsson, and Sadanand where we prove an analogous result with no restrictions on interior angles.  Time permitting, I will also describe a companion result for billiards in hyperbolic polygons with Erlandsson and Sadanand, where the exceptional cases are much more robust.

Short Bio:

After receiving his PhD from the University of Texas at Austin in 2002, Leininger became an NSF Postdoctoral Scholar in the Department of Mathematics at Barnard College and Columbia University in New York.  In 2005, he joined the Mathematics Faculty at the University of Illinois at Urbana Champaign, during which time he was a Helen Corley Petit Scholar (2010-2011) and Lois M. Lackner Scholar (2011-2013).  In 2020, he joined the Department of Mathematics at Rice University and was awarded Rice University's College of Natural Science Breakthrough Research Award in 2025. Leininger was inducted as a Fellow of the American Mathematical Society in 2023.