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.

摘要奖项：DMS-0405578 首席研究员：Christopher Bishop PI Christopher Bishop 将研究共形和拟共形映射的几何性质，重点关注共形结构、双曲几何、低维拓扑和数值分析之间的相互作用。 例如，摩尔定理指出，如果我们在 2-球体上采用特定的集合集合并将它们拓扑折叠为点，那么我们将获得一个新的拓扑球体。 PI 将研究商图何时可以共形，并从这个一般角度考虑许多具体问题，包括共形焊接、John 域的表征、Koebe 猜想、Kleinian 群的构造和其他动态对象。 PI 将继续他早期关于克莱因极限集的几何学和变形极限集时尺寸行为的工作。 PI 还将继续研究计算几何、双曲几何和共形映射之间的联系，并寻求快速计算黎曼映射并具有严格误差估计的新算法。 特别是，他将研究使用中轴（计算几何中的一个对象）计算共形映射，该映射通过凸包与 3 维双曲几何密切相关。共形映射因其在众多数学问题（复杂分析、动力系统等）中的核心作用以及在各种应用（流体流动、脑图绘制、统计物理、数值分析）中的核心作用而非常重要。 微分方程，...），因此我们必须对这些映射有良好的理论理解，并在实践中计算它们的良好方法。该提案通过研究与数学和计算机科学其他部分（点集拓扑、3 维双曲几何、Voronoi 图）的新联系，加深了我们对共形图的理论理解，并寻求利用这些联系来发明计算共形图的新算法。 例如，中轴是计算机科学中广泛研究的对象（它是对对象形状的描述，在模式识别、机器人运动、生物学等领域有大量应用）。 PI 发现它还可以用于对共形图进行粗略但快速的近似。PI 将寻求改进这种方法，为共形图提供更好的算法，并更好地理解中轴，这将影响其其他应用（例如，当所描述的对象仅发生一点变化时，中轴可能会发生巨大变化；这是一个严重的计算问题，可以通过将中轴视为一个对象来解决） 在双曲几何中而不是通常的欧几里得几何中）。中轴和共形映射之间的联系也可能会带来研究三维应用的新想法（共形映射不存在，但基于中轴的映射仍然存在）。三维问题对于应用来说是最重要的，但我们的理解远远落后于二维情况，因此推广到更高维度的新二维思想很重要。