Topics in Projective and Hyperbolic Geometry

建议：DMS-0072607PI：Richard Schwartz摘要：Schwartz建议继续他在以下一般领域的研究：复双曲几何、射影几何中的动力学和计算机辅助数学。第一个主题可以非正式地描述如下。假设一个人在空间中悬挂了有限数量的镜子，并在镜子附近放置了一个物体。透过镜子观察物体，人们可能会看到一个无限的、有规律的图案，就像橘子树林里的树一样。另一方面，人们可能会看到一种混乱和混乱的模式，充满了部分和重叠的图像。第一种情况大致对应于S所说的离散群，第二种情况大致对应于所谓的非离散群。人们想要研究的基本问题是：镜子的哪些位置通向离散的替代方案？在施瓦茨的研究中，悬挂镜子的空间是一个被称为复双曲空间的弯曲的4维宇宙。这个空间是一百多年前由Gauss和Lobachevsky构造的著名的非欧几里德空间的异国表亲。第二个主题涉及直线几何的简单构造。射影几何中的经典定理，如巴普斯定理和德沙格定理，有时可以反复应用，而不是只应用一次。其结果是一种动力系统，涉及一个无穷无尽的点和线家族。动力系统背后的数学通常超越了原始结果背后的数学。例如，Schwartz研究的一个例子导致了与可积偏微分方程式、行列式恒等式和交替符号矩阵的联系。施瓦茨建议继续研究这些动力系统。第三个主题涉及计算机辅助数学。人们经常会遇到这样一种情况，即计算机说某个结果是真的，但证据无从洞察。施瓦茨计划调查几种情况，在这些情况下，可以直接使用计算机的输出作为证明结果为真的基础。换句话说，计算本身就成了结果的正当性。更具体地说，施瓦茨想要尝试根据轨道上有限数量的信息来推断某些类型的动力系统的整个轨道结构。显然，这样的目标只有在特殊情况下才会奏效。

Proposal: DMS-0072607PI: Richard SchwartzAbstract: 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.