Topics in Projective and Hyperbolic Geometry

摘要：Schwartz建议在复双曲几何、射影几何动力学和计算机辅助数学等领域继续他的研究。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有限数量的镜子，并在镜子附近放置一个物体。透过镜子看这个物体，人们可能会看到一个无限的、规则的图案，就像橘子林中的树一样。另一方面，人们可能会看到一个混乱和混乱的图案，充满了局部和重叠的图像。第一种情况大致对应于所谓的离散群，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：镜子的哪些位置会导致离散的选择？在施瓦茨的研究中，悬挂镜子的空间是一个弯曲的四维宇宙，称为复双曲空间。这个空间是高斯和罗巴切夫斯基一百多年前构造的著名的非欧几里得空间的异域表亲。第二个主题涉及直线几何中的简单构造。射影几何中的经典定理，如帕普斯定理和德萨格定理，有时可以反复应用，而不仅仅是一次。其结果是一种动力系统，涉及无穷一族的点和线。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，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的情况：计算机说某个结果是正确的，但证明却无处可寻。施瓦茨计划研究几种情况，在这些情况下，可能直接使用计算机的输出作为证明结果正确的基础。换句话说，计算本身成为对结果的证明。更具体地说，施瓦茨想要根据轨道上有限的信息，尝试推断出某些动力系统的整个轨道结构。显然，这样的目标只会在特殊情况下起作用。

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.