DescriptionIn this project we will study properties of surfaces of constant negative curvature. The universal covering of such a surface is the hyperbolic plane, and the beauty of hyperbolic geometry is that one can use techniques of many different mathematical disciplines like Differential and Riemannian Geometry, Algebra and Number Theory, Analysis and Dynamical Systems.A natural geometric dynamical system on hyperbolic surfaces is the geodesic flow, which has both a geometric and an algebraic description. Of particular interest in dynamical systems are closed trajectories, and a nice feature of closed geodesics on hyperbolic surfaces is that they have a length. If one restricts to special hyperbolic surfaces (like the modular surface related to the subgroup SL(2,Z) or SL(2,R)), then the study of closed geodesics and their length leads naturally to a lot of algebra and number theory (continuous fractions, best rational approximation of irrational numbers, ideals in real quadratic number fields, class numbers, discriminants and regulars). The sequence of lengths of closed geodesics of a hyperbolic surface is called its length spectrum. Analysis enters into the game via the Laplace operator of a hyperbolic surface. In the classical setting, the eigenvalues of the Laplacian are related to frequencies of a drum with the corresponding shape. The Laplacian provided another sequence of data: its eigenvalues, which one calls the Laplace spectrum. In contrast to the length spectrum, which is related to traces of a certain subgroup of matrices in SL(2,R), the Laplace spectrum, however, is very difficult to calculate explicitely. There are beautiful connections between these two sets of numbers, the length and the Laplace spectrum, culminating in the famous Selberg's trace formula. This project allows development in a variety of directions reflecting the background of the students. It is necessary that the students have some basic knowledge in hyperbolic geometry, so they must have taken one of the courses Geometry III/IV or Differential Geometry III. Some ResourcesThe books
email: N Peyerimhoff |