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.
