Deformations of Complex Structures

复杂结构的变形

• 批准号: 9800924
• 负责人: Christopher Bishop
• 金额: $ 23.48万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800924&HistoricalAwards=false
• 关键词: Deformations; Complex; Structures

Prof. Bishop will investigate several aspects of geometry and analysis related to non-smooth objects such as limit sets of Kleinian groups, Brownian motion and quasi-conformal mappings. Among the specific goals are to extend his earlier application of harmonic analysis and heat kernel techniques to compute the exact size (in terms of Hausdorff measures) of the limit sets of Kleinian groups to new examples (e.g., manifolds with thin parts) and new sets (the conical limit set, the set of escaping geodesics,...). He will also attempt to extend Bowen's famous dichotomy to all divergence type Fuchsian groups, extending work of Sullivan, Astala, Zinsmeister, Jones and himself. Part of this is to further study the ``beta''s, a technical measure of how close a set is to a line segment at different scales, which is closely related to the study of the traveling salesman problem. He will also continue his work investigating the removable and non-removable sets for quasi-conformal mappings, geometric properties of Brownian motion, and other geometric-analytic questions in classical complex analysis.The proposal deals with a variety of geometric questions which are united by a desire to understand traditional concepts and calculations (e.g. compute the area of a region) in non-traditional and highly non-smooth cases. This is motivated by (at least) two points of view. First, a better understanding of the classical case can be obtained by seeing how it succeeds or fails in a more general setting; even the failures lead to new, interesting phenomena to investigate. Second, many applications of mathematics (polymers, fractures, shock waves, wavelet analysis, crystal growth,...) involve highly irregular and fractal objects, and we need to understand and calculate with such objects the way we have always done with much smoother quantities. For example, Brownian motion is a mathematical object which models random movement and is of tremendous intrinsic interest, as well as being fundamental to understanding many other random processes (e.g., shapes of long polymers chains, growth of crystals by random accretion, ...). However, Brownian motion is an extremely non-smooth process and various simple geometric questions about its random paths are still unknown. Another example is with the Kleinian groups mentioned above. These are symmetries in non-Euclidean geometry; as such, they are fundamental objects in topology and have been an area of intense investigation for many years. They can be associated with certain fractal objects (the limit sets) involving non-linear rescalings, where the non-linearity is of a very special type, and present a theoretical stepping stone between self-similar sets with linear rescalings (which are fairly well understood) and much more complicated rescalings found in rational dynamics and chaos theory.

Bishop教授将研究与非平滑物体有关的几何形状和分析的几个方面，例如克莱尼亚组的极限集，布朗运动和准符合形式映射。 在具体目标中，他的早期应用谐波分析和加热内核技术以计算Kleinian群体的极限集的确切大小（根据Hausdorff度量）的确切大小（例如，较薄的零件的流形）和新集合（锥形限制集合，逃避地理学的集合，...）。 他还将试图将鲍恩（Bowen）著名的二分法延伸到所有分歧型富奇族（Fuchsian type Fluchsian）团体，扩展了沙利文（Sullivan），阿斯塔拉（Astala），Zinsmeister，琼斯（Jones）和他本人的作品。其中的一部分是进一步研究``beta''的技术，这是一种技术衡量，以衡量一组在不同尺度上的线段的距离，这与对旅行推销员问题的研究密切相关。 他还将继续他的工作调查，以用于准正形映射，布朗尼运动的几何特性以及在古典复合物分析中的其他几何分析问题的可拆卸和不可移动的集合。该提案涉及多种几何问题，这些问题涉及各种愿望了解传统概念和计算的愿望（例如，都不理解一个区域和不合格的区域）。 这是（至少）两种观点的动机。 首先，可以通过查看在更一般的环境中如何成功或失败来更好地理解经典案例；甚至失败也会导致新的有趣现象进行调查。 其次，数学（聚合物，断裂，冲击波，小波分析，晶体生长等）的许多应用都涉及高度不规则和分形对象，我们需要按照我们始终以多种更柔和的数量来理解和计算此类物体。 例如，布朗运动是一个数学对象，它模拟随机运动，并且具有巨大的内在兴趣，并且是理解许多其他随机过程的基础（例如，长聚合物链的形状，通过随机积聚的晶体生长，...）。但是，布朗运动是一个极为平滑的过程，关于其随机路径的各种简单几何问题仍然未知。 另一个例子是上面提到的kleinian群体。 这些是非欧盟几何形状中的对称性；因此，它们是拓扑中的基本对象，并且多年来一直是深入研究的领域。 它们可以与涉及非线性重塑的某些分形对象（极限集）相关联，在这些分类物体中，非线性具有非常特殊的类型，并在具有线性恢复的自相似套件之间存在理论上的步进石材（相当理解），并且在理性动力学和Chaos理论中发现了更为复杂的重新分类。