On Fuchsian Groups and Fundamental Domains of Outer Billiards Background: Orientation preserving isometries of the hyperbolic plane are classified as elliptic, parabolic or hyperbolic, depending on the number and location of their fixed points: an elliptic isometry has a unique fixed point in the hyperbolic plane (and therefore is a rotation); a parabolic one has a unique fixed point on the circle at infinity; and a hyperbolic one has two fixed points on the circle at infinity, one attracting and one repelling, both of hyperbolic type. A discrete subgroup of isometries of the hyperbolic plane is called a Fuchsian group if it consists of orientation-preserving transformations. Any Fuchsian group possesses connected, convex fundamental domain. 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. 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. We are going to study the map in the hyperbolic plane, using mostly Klein or Poincare models (two common models for hyperbolic geometry). Previous Undergraduate Research Experiences: I have mentored seven undergraduate students. In these research programs my students worked on geometry and orbit structures of the outer billiard map in the hyperbolic plane. 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. REU - 2016: When the outer billiard table is a regular n-gon (n > 4) with all right angles, the dynamics of corresponding outer map is already analyzed. This summer students will consider the dynamics of the map corresponding different tiling table and we are going to explore fundamental domains of outer billiard map defined with non-tiling polygonal tables and smooth tables in the hyperbolic plane and/or Euclidean plane. 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.