Billiards & Outer (Dual) Billiards Background: Outer (dual) billiard dynamical systems belongs to a major mathematical discipline, the dynamical systems, which studies the orbit structure of maps and flows. The outer billiard map is an outer counterpart of the usual billiard map. The theory of dynamical systems of billiard type has been a very popular object of study in mathematics, mechanics and physics. Mathematicians and physicists have studied geometrical optics which is intimately related to billiards. One may say that every mechanical system with elastic collision can be described as a certain billiard. An outer billiard is a simple dynamical system in the plane based on a convex shape. B.H. Neumann introduced outer billiards in the 1950s and J. Moser popularized it in the 1970s. The main and interesting question for the map is “Is it possible for orbits to escape infinity?” In time, several mathematicians provided partial answers for one particular class of polygons called quasirational, in the Euclidean plane. However, the complete answer is still not known. Definition: Formally, given a compact convex plane domain P (outer billiard table), one defines the outer billiard transformation F of its exterior as a reflection through a support point in the given direction. This definition applies if the support point is unique; otherwise the transformation is not defined. I study the map in the hyperbolic plane, using mostly Klein or Poincare models (two common models for hyperbolic geometry). My first and significant contribution to the field was finding a class of polygons for which the outer billiard orbit escapes to infinity in the hyperbolic plane. I also analyzed the dynamics of the outer billiard map when the dual billiard is a regular n-gon (n > 4) with all right angles. Additionally, I characterized the derivative of the extended dual billiard map at periodic points. Previous Undergraduate Research Experiences: I have mentored three undergraduate students. In these research programs my students worked on geometry and orbit structures of the outer billiard map in the hyperbolic plane as well. As a result of these collaborations, two papers was published, one other is submitted. Each one of my students presented their work in several conferences and each one them received different awards and prizes. 2013 REU : This summer students are going to analyze the extended outer (dual ) billiard map at the infinity of hyperbolic planes. It is a circle map. We are going to explore rational and irrational rotation numbers of the map with respect to different tables. We are going to use Klein, Poincare and Upper Half Plane models. Desirable Experience For Applicants: Students are expected to have Calculus, Linear Algebra, and Euclidean background. Strong programing skills and Non-Euclidean Geometry background will help us to extend the research to the next step. However, students without programing experience with good mathematical backgrounds are encouraged to apply and will also be closely considered. How To Apply: For application information and instructions, please visit the GVSU Summer Mathematics REU home page.