Geometry of Conformal and Quasiconformal Mappings

摘要DMS -0103626的PI，克里斯托弗·毕晓普，将研究的几何性质的共形映射在平面和拟共形映射在空间中，侧重于这种映射的扩展性质和阐述各种应用几何函数理论，动力学和拓扑。 PI已经表明，结果丹尼斯沙利文's关于几何的凸体在双曲三空间意味着一个factorizationtheorem共形映射在平面上，这反过来又意味着统一的界限上的数额ofconstraction共形映射在平面上可以有。 寻找最佳常数的因式分解定理的后果众所周知的问题，如维数失真，积分手段和布伦南猜想。 PI将继续研究Kleinian群的极限集，Kleinian群是分形集的一个自然而重要的类别，这里的问题主要是估计这些集合的分维，并研究当群变形时维数的行为。PI还将研究调和测度的度量性质，特别是量化调和测度不能集中在一个小集合上的思想的结果。 问题包括低密度猜想、调和测度的稳定性和扩散限制聚集的增长率。 共形映射是一类在数学的许多领域都很重要的函数，它与许多物理问题（流体流动、热传导、电场、随机增长模型等）密切相关。并且已经被深入研究多年。 这类地图的基本性质之一是膨胀;它们往往会使点平均距离更远。 使这一精确激发了数学分析中的许多研究。PI发现了一种新的方法来量化这种扩展，通过用（更一般的）拟共形映射来近似共形映射，并显示这些近似可能具有非常强的扩展性质。 这使我们对一些已知的结果有了更清楚的理解，并在新问题上取得了进展，特别是，这意味着关于Kleinian群的新结果（这些是共形动力系统的重要例子，因此对动力系统、分形和混沌等更广泛的领域做出了贡献）。 PI的方法也将保形映射的行为与三维双曲空间的几何联系起来;这种联系似乎是新的，应该会导致许多有趣的问题，以及复分析和三维拓扑（已经以其他方式连接）领域之间的更多互动。 他还将研究这种连接的计算方面，这可能会导致计算保形映射和格林函数（重要的各种应用）的新方法。 PI还将继续他的调查其他问题，包括几何随机路径，如布朗运动，稳定性扰动下的某些动力系统和fundamentalgeometric性质的共形和quasiconformal映射。

Abstract for DMS - 0103626The PI, Christopher Bishop, will study the geometric propertiesof conformal mappings in the plane and quasiconformal mappings in space,focusing on the expansion properties of such maps and investigatingvarious applications to geometric function theory, dynamics andtopology. The PI has shown that a result of Dennis Sullivan'sconcerning the geometry of convexbodies in hyperbolic three space implies a factorizationtheorem for conformal mappings in the plane and this, in turn,implies uniform bounds on the amount ofcontraction a conformal map in the plane can have. Finding the best constantsin the factorization theorem has consequences for wellknown problems such as dimension distortion, integral means and Brennan'sconjecture. The PI will continue his work onlimit sets of Kleinian groups, a natural and important class of fractal sets.The questions here are mainly to estimate the fractal dimension of these setsand study the behavior of the dimension as the group is deformed. The PI will also work on the metric propertiesof harmonic measures, particularly results which quantify the idea that harmonicmeasure cannot be concentrated on a small set. Problems include thelower density conjecture, stability of harmonic measure and the growth rateof diffusion limited aggregation. A few other questionsinvolving quasiconformal and biLipschitz maps are also considered.Conformal mappings are a class of functions which are importantin many area of mathematics and which are closely relatedto mnay physical problems (fluid flow, heat conduction, electric fields, random growth models, ...) and have been intensively studied for many years. One of the fundamental properties of such maps is expansion; they tend to push points farther apart on average. Making this precise has motivated much research in mathematical analysis.The PI has discovered a new way of quantifyingthis expansion by approximating conformal maps by (the moregeneral class of) quasiconformal maps and showing theseapproximations may be taken with a very strong expansion property. This has given a clearer understanding of someknown results and has led to progress on new problems.In particular, it implies new results about Kleinian groups(these are important examples of conformal dynamical systems, and hence a contribution to the more general area of dynamical systems, fractals and chaos). The PI's approach also ties the behavior of conformal maps to the geometry three dimensional hyperbolic space; this connection seems to be new and should lead to many interesting problems and more interaction between the areas of complex analysis and three dimensional topology (already connected in other ways). He will also investigate the computationalaspects of this connection which may lead to new methods of computing conformal maps and Greens functions (importantfor a variety of applications). The PI will also continue his investigation of other problems including the geometry of randompaths such as Brownian motion, the stability underperturbation of certain dynamical systems and fundamentalgeometric properties of conformal and quasiconformal mappings.