Geometry and Dynamics

9803526施瓦茨理查德·施瓦茨将沿着四条路线开展几何学和动力学研究。首先，Schwartz将继续他在1993年证明的非均匀一阶格是准等距的当且仅当它们是可公度的。特别是，施瓦茨将进一步发展行动刚性的主题，这一主题在他1996年的ACTA论文《准等距刚性和丢番图近似》中被引入。其次，施瓦茨将探索他1997年证明戈德曼-帕克猜想的结果，该猜想给出了复杂双曲理想三角群的精确离散性标准。这是第一个这样的结果，因为它给出了复双曲空间中的一个非平凡变形问题的完整分析。第三，施瓦茨将继续他在射影几何中迭代构造所定义的动力系统方面的工作。这里的工作涉及到双星动力学、阿贝尔变种、KdV方程和行列式。第四，施瓦茨将开发帮助理解、交流和解决几何和动力学问题的软件。这项研究的共同主题是，它处理无限的、对称的图案的几何结构。例如，人们应该想到厨房地板上的瓷砖，或者橘子树林里排列的树木。非正式地说，施瓦茨研究的第一个领域涉及这样一个问题：一个相当盲人会把一种无限对称模式误认为另一种吗？我们的想法是找到无限模式中非常“粗略”的特征，以将其与另一种无限模式区分开来。第二个研究领域涉及以下几类问题：想象一个苹果悬浮在太空中，周围环绕着镜子。你怎么才能把镜子放在合适的位置，让苹果的反射图案美观有序？第三个研究领域涉及通过重复简单的结构来产生无限的图案，这些结构由铅笔和纸画制成。通常一个非常无辜的结构，如帕普斯的定理结构，在无限重复的情况下会产生一个令人惊讶和美丽的几何图案。施瓦茨第四领域研究的目的是通过交互计算机图形学使所有这些无限的模式变得生动起来。***

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. ***