Deformations of Complex Structures

Bishop教授将研究与非光滑对象相关的几何和分析的几个方面，如Kleinian群的极限集，布朗运动和准共形映射。 具体目标之一是将他早期应用调和分析和热核技术来计算Kleinian群极限集的精确大小（根据Hausdorff测度）扩展到新的例子（例如，流形薄部分）和新的集（圆锥极限集，逃逸测地线集，.）。 他还将尝试将鲍文著名的二分法扩展到所有发散型Fuchsian群体，扩展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”，这是一种技术措施，衡量一个集合在不同尺度下与线段的接近程度，这与旅行商问题的研究密切相关。 他还将继续他的工作调查的可移动和不可移动集的拟共形映射，布朗运动的几何性质，以及其他几何分析问题在经典的复analysis.The建议涉及各种几何问题，这是由一个愿望，以了解传统的概念和计算（例如计算区域的面积）在非传统和高度非光滑的情况下团结。 这是由（至少）两个观点引起的。 首先，通过观察经典案例在更一般的环境中是如何成功或失败的，可以更好地理解经典案例;即使失败也会导致新的、有趣的现象。 第二，数学的许多应用（聚合物、断裂、冲击波、小波分析、晶体生长等）涉及高度不规则和分形的物体，我们需要理解和计算这些物体，就像我们一直用更平滑的量做的那样。 例如，布朗运动是一种数学对象，它对随机运动进行建模，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，通过随机增长的晶体生长，.）。然而，布朗运动是一个极其非光滑的过程，关于它的随机路径的各种简单几何问题仍然是未知的。 另一个例子是上面提到的Kleinian群。 这些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，多年来一直是一个激烈的研究领域。 它们可以与涉及非线性重标度的某些分形对象（极限集）相关联，其中非线性是一种非常特殊的类型，并且在具有线性重标度的自相似集（这是相当好理解的）和在理性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.