Geometry of Conformal and Quasiconformal Mappings

共形和拟共形映射的几何

• 批准号: 0405578
• 负责人: Christopher Bishop
• 金额: $ 20万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-15 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405578&HistoricalAwards=false
• 关键词: Geometry; Conformal; Quasiconformal; Mappings

AbstractAward: DMS-0405578Principal Investigator: Christopher BishopThe PI, Christopher Bishop, will study the geometric propertiesof conformal and quasiconformal mappings, focusing on theinteractions between conformal structures, hyperbolic geometry,low dimensional topology and numerical analysis. For example,Moore's theorem states that if we take a certain collection ofsets on the 2-sphere and topologically collapse them to pointsthen we obtain a new topological sphere. The PI will investigatewhen the quotient map can be conformal and will consider a numberof concrete problems from this general perspective, includingconformal welding, characterizations of John domains, Koebe'sconjecture, construction of Kleinian groups and other dynamicalobjects. The PI will continue his earlier work on the geometryof Kleinian limit sets and the behavior of the dimension as wedeform the limit set. The PI will also continue his work on theconnections between computational geometry, hyperbolic geometryand conformal mappings, and seek new algorithms which compute theRiemann mapping quickly and with rigorous error estimates. Inparticular he will investigate computing conformal maps using themedial axis (an object from computational geometry) which isclosely linked to 3-dimensional hyperbolic geometry via convexhulls.Conformal mappings are important both for their central role innumerous mathematical problems (complex analysis, dynamicalsystems,...) and in various applications (fluid flow, brainmapping, statistical physics, numerical analysis of differentialequations,...), so we must have a good theoretical understandingof these maps and good methods for computing them inpractice. The proposal deepens our theoretical understanding ofconformal maps by investigating new connections with other partsof mathematics and computer science (point set topology,3-dimensional hyperbolic geometry, Voronoi diagrams) and seeks touse these connections to invent new algorithms for computingconformal maps. For example, the medial axis is a widely studiedobject in computer science (it is a description of the shape onan object which has numerous applications in pattern recognition,robotic motion, biology,...). The PI discovered it can also beused to give a rough but fast approximation to conformal maps.The PI will seek to improve this method, giving better algorithmsfor conformal maps and also developing a better understanding ofthe medial axis which will impact its other applications (forexample, the medial axis can change drastically when the objectbeing described changes only a little; this is a seriouscomputational problem which can be addressed by thinking of themedial axis as an object in hyperbolic geometry instead of theusual Euclidean geometry). The connection between the medial axisand conformal maps may also lead to new ideas for studyingapplications in three dimensions (where conformal mappings do notexist, but maps based on the medial axis still do). Threedimensional problems are the most important for applications, butour understanding lags far behind the two dimensional case, sonew two dimensional ideas which generalize to higher dimensionsare important.

Abstractaward：DMS-0405578原理研究者：Christopher Bishopthe Pi，Christopher Bishop将研究保形和准文献映射的几何特性，重点介绍了整形结构，多型几何学，多型拓扑和数字分析之间的相互作用。 例如，摩尔的定理指出，如果我们在2个球员上采取一定的集合，然后拓扑崩溃，然后将其倒入点，然后将获得一个新的拓扑领域。 PI将研究商映射可以是保形的，并将从这个一般角度考虑一个数字混凝土问题，包括符号焊接，John域的特征，Koebe' -Sconjecture，Kleinian组的构建和其他动态对象。 PI将继续他对Kleinian限制集的几何形状以及维数的行为的较早工作。 PI还将继续他在计算几何形状，双曲线几何和共形映射之间的连接方面的工作，并寻求新的算法，这些算法可以快速计算theriemann映射，并进行严格的错误估计。 他将使用主题轴（计算几何形状的对象）研究计算结构图，该轴通过凸面与三维双波利几何形状链接到3维的双层几何形状。构型映射对其中心作用既重要，这对于它们的核心作用无关紧要（复杂的分析，动力学分析，动力学，脑，脑，脑，脑，脑，脑，脑含量不多）差异化，...），因此我们必须对这些地图的理论了解良好，并有良好的方法来计算它们的实践。该提案通过调查与数学和计算机科学的其他部分（点集拓扑，三维双曲几何形状，Voronoi图）的新联系来加深我们对整形图的理论理解，并寻求这些连接以发明用于计算概念图的新算法。 例如，内侧轴是计算机科学中广泛研究的对象（这是对形状Onan对象的描述，该对象在模式识别，机器人运动，生物学等中都有许多应用。 PI发现的PI也可能会构成一个粗糙但快速近似的形式。PI将寻求改进该方法，提供更好的算法来提供保质图，并发展对内侧轴的更好理解，这将影响其其他应用，这将影响其其他应用（forampample）（tofamplame forixame），当对象范围的问题上可以改变一定的问题，这是一个严重的问题。在双曲几何形状而不是尤克里亚人的几何形状中）。内侧轴和共形图之间的连接也可能导致在三个维度上进行研究的新想法（在三个维度上进行研究（共形映射确实不是验证，但基于内侧轴的地图仍然可以做到）。三维问题对于应用是最重要的，但是了解远远落后于二维案例的滞后，即二维思想，这些思想推广到更高的维度。