Geometry and Dynamics

Richard Schwartz将沿着四条路线进行几何学和动力学方面的研究。首先，Schwartz将继续他1993年开始的工作，证明非均匀秩一格是准等距的当且仅当它们是可通约的。特别是，Schwartz将进一步发展他在1996年的Acta论文“准等距刚性和丢番图近似”中介绍的作用刚性主题。其次，施瓦茨将探索他1997年证明戈德曼-帕克猜想的分支，该猜想给出了复杂双曲理想三角形群的精确离散准则。该结果首次完整地分析了复双曲空间中的非平凡变形问题。第三，Schwartz将继续他在射影几何中由迭代构造定义的动力系统方面的工作。这里的工作涉及到双民族动力学，阿贝尔变种，KdV方程和行列式。第四，施瓦茨将开发有助于理解、交流和解决几何和动态问题的软件。这项研究的共同主题是它处理无限对称图案的几何结构。例如，人们应该想到厨房地板上的瓷砖，或者橘子林中排列的树木。非正式地说，施瓦茨研究的第一个领域涉及的问题是：一个相当失明的人会把一种无限对称模式误认为另一种吗？这个想法是找到一个无限模式的非常“粗略”的特征，可以将它与另一个无限模式区分开来。第二个研究领域涉及以下问题：想象一个苹果悬浮在太空中，四周都是镜子。你如何放置镜子，使苹果的反射图案美观有序？第三个研究领域涉及通过铅笔和纸画的简单结构的重复来产生无限的图案。通常，一个非常单纯的结构，如帕普斯定理结构，在无限重复后会产生令人惊讶而美丽的几何图案。Schwartz研究的第四个领域的目的是通过交互式计算机图形使所有这些无限的模式变得生动。＊＊＊

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. ***