Topics in Projective and Hyperbolic Geometry

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Proposal: DMS-0072607 PI: Richard Schwartz Abstract: Schwartz proposes to continue his research in the following general areas: Complex Hyperbolic Geometry, Dynamics in Projective Geometry, and Computer-Aided Mathematics. The first topic can be described informally as follows. Suppose one suspends a finite number of mirrors in space, and places an object in the vicinity of the mirrors. Looking at the object through the mirrors, one might see an infinite, regular pattern, like trees in an orange grove. On the other hand, one might see a confused and chaotic pattern, full of partial and overlapping images. The first case corresponds roughly to what s called a discrete group and the second corresponds roughly to what is called an indiscrete group. The basic question one would like to study is: which positions of the mirrors lead to the discrete alternative? In the case of Schwartz's research, the space in which the mirrors are suspended is a curved 4-dimensionaluniverse called complex hyperbolic space. This space is an exotic cousin of the famous non-Euclidean spaces constructed by Gauss and Lobachevsky more than a hundred years ago. The second topic involves simple constructions in straight-line geometry. The classical theorems in projective geometry, such as Pappus's theorem and Desargues theorem, can sometimes be applied over and over again, rather than just once. The result is a kind of dynamical system, involving an infinite family of points and lines. The mathematics behind the dynamical system usually transcends the mathematics behind the original result. For instance, one example studied by Schwartz leads to connections with integrable partial differential equations, determinental identities, and alternating sign matrices. Schwartz proposes to continue investigating these dynamical systems. The third topic involves computer aided mathematics. One frequently encounters a situation where the computer says that a certain result is true, but a proof is nowhere insight. Schwartz plans to investigate several situations where it might be possible to use the output of the computer directly as the basis of a proof that the result is true. In other words, the computation itself becomes the justification of the result. More concretely, Schwartz would like to try to deduce the entire orbit structure of certain kinds of dynamical systems based on a finite amount of information on the orbit. Obviously such a goal would only work in special situations.