Geometry and Dynamics

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9803526 Schwartz Richard Schwartz will carry out research in geometry and dynamics, along four lines. First, Schwartz will continue the work that began with his proof, in 1993, that nonuniform rank one lattices are quasi-isometric if and only if they are commensurable. In particular, Schwartz will further develop the theme of action rigidity, introduced in his 1996 Acta paper, "Quasi-isometric rigidity and diophantine approximation." Second, Schwartz will explore the ramifications of his 1997 proof of the Goldman-Parker Conjecture, which gives exact discreteness criteria for complex hyperbolic ideal triangle groups. This result is the first of its kind, in that it gives the complete analysis of a nontrivial deformation problem in complex hyperbolic space. Third, Schwartz will continue his work on dynamical systems defined by iterative constructions in projective geometry. The work here is related to birational dynamics, abelian varieties, the KdV equation, and determinants. Fourth, Schwartz will develop software that aids in the understanding, communication, and solution of geometric and dynamical problems. The common theme to this research is that it deals with the geometric structure of infinite, symmetric, patterns. One should think, for example, of tiles on the kitchen floor, or trees arrayed in an orange grove. Informally speaking, the first area of Schwartz' research deals with the question: Could a fairly blind person mistake one infinite symmetric pattern for another? The idea is to find very "rough" features of an infinite pattern that could distinguish it from another infinite pattern. The second area of research deals with the following kinds of questions: Imagine that an apple is suspended in space, surrounded by mirrors. How can you position the mirrors so that the pattern of reflections of the apple will be nice and orderly? The third area of research deals with the generation of infinite patterns through the repetition of simpl e constructions, made from pencil and paper drawings. Often a very innocent construction, such as the Pappus' theorem construction, yields a surprising and beautiful geometric pattern upon infinite repetition. The purpose of Schwartz fourth area of research is to make all these infinite patterns come alive through interactive computer graphics. ***