Deformations of Complex Structures

毕晓普教授将研究与非光滑对象有关的几何和分析的几个方面，如克莱因群的极限集、布朗运动和拟共形映射。其中的具体目标是将他早期应用调和分析和热核技术来计算Klein群极限集的精确大小(以Hausdorff度量为单位)扩展到新的例子(例如，具有薄部分的流形)和新的集合(圆锥极限集、逃逸测地线的集合，等等)。他还将尝试将Bowen著名的二分法推广到所有发散型Fuchsian群，推广Sullivan、Astala、Zinsmeister、Jones和他自己的工作。其中一部分是为了进一步研究“贝塔”S，这是一种衡量集合在不同尺度上与线段的距离的技术度量，它与旅行商问题的研究密切相关。他还将继续他的工作，研究准共形映射的可去集和不可去集，布朗运动的几何性质，以及经典复分析中的其他几何分析问题。该建议处理各种几何问题，这些问题是由于希望理解非传统和高度非光滑情况下的传统概念和计算(例如计算区域面积)而结合在一起的。这是由(至少)两个观点推动的。首先，通过在更一般的环境中观察经典案例的成功或失败，可以更好地理解它；即使是失败，也会导致新的、有趣的现象需要研究。第二，数学的许多应用(聚合物、断裂、冲击波、小波分析、晶体生长……)涉及高度不规则和分形的物体，我们需要理解和计算这类物体，就像我们总是用更平滑的量来做的那样。例如，布朗运动是一个数学对象，它模拟随机运动，具有巨大的内在兴趣，也是理解许多其他随机过程的基础(例如，长聚合物链的形状，通过随机吸积生长晶体等)。然而，布朗运动是一个极其非光滑的过程，关于其随机路径的各种简单几何问题仍是未知的。另一个例子是上面提到的克莱恩集团。它们是非欧几里德几何中的对称性；因此，它们是拓扑学中的基本对象，多年来一直是一个密集研究的领域。它们可以与某些涉及非线性重标度的分形对象(极限集)联系在一起，其中非线性是一种非常特殊的类型，并在具有线性重标度的自相似集(这是相当好地理解的)和在有理动力学和混沌理论中发现的更复杂的重标度之间提供了理论上的垫脚石。

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.