# Department of Mathematics

# Meet Pavel Javornik!

Meet one of our undergraduates and summer research interns, Pavel Javornik!

This summer I will be continuing my work under the mentorship of Dr. Patrick Hooper. Our current project is to describe the dynamical properties of geodesic flows on non-compact surfaces composed entirely of boundary unions of various polyhedrons. Certain characteristics of these infinite surfaces, such as the symmetries of the canonical forms of their quotient spaces, determine the behavior of geodesic flows given properties of said flow (such as their initial trajectories). The goal this summer is to adapt various methods used in studying infinite surfaces constructed from compact translation surfaces to better understand surfaces that might admit transformations of geodesic flows in the form of rotations. Much of the study of Veech surfaces is applicable to certain rational billiards problems on infinite surfaces like the Ehrenfest-Wind Tree Model, but these transformations often admit reflections off of boundaries in the form of perfectly elastic collisions. The translation surfaces and (consequently) Veech groups of these infinite surfaces have symmetrical properties unlike those of surfaces constructed of polyhedrons. Understanding how these boundaries might affect geodesic flow on flattened structures is key to understanding their dynamical properties.

What drives my work is my love for mathematics. Studying the underlying structures of objects such as manifolds fascinates me. With low-dimensional topology there is a geometric intuition when trying to characterize these kinds of surfaces. Describing the homology classes of non-compact, infinite (possibly infinite genus) surfaces in the form of their compact covering/translation spaces is a somewhat novel undertaking. There's an extraordinary number of possibilities in this realm of mathematics and they all begin with asking simple questions that begin to unravel the mysteries of the objects we study.

Posted 06/14/2017; expires 07/14/2018; visible to public

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