Mathematical Sciences: Discrete Groups and Quasiconformal Mappings

数学科学：离散群和拟共形映射

• 批准号: 9622808
• 负责人: Frederick Gehring
• 金额: $ 4.79万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622808&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Discrete; Groups; Quasiconformal

ABSTRACT Proposal: DMS-9622808 PI: Gehring This project addresses problems concerning discrete groups of Mobius transformations, 3-manifolds and orbifolds, quasiconformal mappings, and function theoretic properties of plane domains. For example, the Gehring plans to continue a joint study with G. J. Martin of the set of values which cannot be assumed by the trace of the commutator of pairs of elements of a nonelementary discrete group. This study has already yielded a great deal of information about the minimum possible volumes of 3-manifolds and orbifolds and it should, in particular, allow one to prove that the orbifold associated with the 3-5-3 hyperbolic tetrahedral group has minimum volume among all 3-manifolds and orbifolds. The main tools for this study are: (1) complex iteration to identify big filled in Julia sets E for a special family P of polynomials, and (2) a covering argument using the family P and the filled in Julia sets E to obtain large regions D of excluded values for the traces of the commutators of pairs of elements. The Gehring hopes also to study further the polynomial family P, an interesting collection of polynomials with one complex parameter which arise naturally in the study of Mobius groups. Mathematics can be viewed as consisting of three main fields - algebra, analysis, and geometry - together with many other areas which are developments of the basic ideas in one or more of these fields. Thus algebra is the field which has developed from studying polynomial equations, analysis the field which has its roots in the calculus, and geometry the field that has developed from the ideas of Euclid and the Greeks. It is particularly interesting and satisfying for mathematicians when ideas which belong to one of these fields can be used to solve problems in another. This project concerns some problems which are situated at a crossroads where all three of these fields meet. In particular, one problem is to determine the smallest possible volume of certain g eometric objects using the theory of complex iteration and Mandelbrot sets of analysis together with a polynomial family which arose from algebra. Work in such a part of mathematics can be quite fruitful since the solution of a crossroads problem usually has implications in all of the fields concerned.

摘要建议：DMS-9622808 PI：GEHRING该项目解决了有关Mobius转换，3个manifolds和Orbifolds，Arbifolds，Quasiconformal映射以及平面域的功能理论特性的问题。 例如，Gehring计划继续与G. J. Martin进行联合研究，该研究值是由非元素离散组的成对元素的轨迹所无法承担的。 这项研究已经产生了有关3个manifolds和Orbifolds可能最小可能体积的大量信息，特别是，它应该允许人们证明与3-5-3双曲四面体组相关的Orbifold在所有3- manifolds和Orbifolds中具有最小的四个倍数。这项研究的主要工具是：（1）复杂的迭代，以确定特殊多项式家庭的朱莉娅集合E中的大量填充，以及（2）使用family P和填充在Julia集合中的涵盖论点，以获得元素对换对元素的痕迹的巨大区域d。 Gehring也希望进一步研究多项式家族P，这是一个有趣的多项式集合，其中一个复杂参数是在Mobius组研究中自然出现的。 数学可以看作是由三个主要领域（代数，分析和几何形状）组成，以及许多其他领域，这些领域是这些领域中一个或多个领域中基本思想的发展。 因此，代数是从研究多项式方程，分析其根源在微积分中的领域以及从欧几里得和希腊人的思想发展的田地发展的领域。 当属于这些领域之一的想法可以用来解决另一个领域的问题时，这对数学家特别有趣且令人满意。 该项目涉及一些位于这三个领域相遇的十字路口的问题。 特别是，一个问题是使用复杂的迭代和曼德布罗特分析的理论以及来自代数的多项式家族来确定某些g子代对象的最小体积。在此类数学的一部分中的工作可能会非常富有成果，因为解决问题的解决方案通常对所有有关的领域都有影响。