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首席研究员：Christopher Bishop首席研究员Christopher Bishop将研究共形和拟共形映射的几何性质，重点是共形结构，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-球面上取一定的集合，并将它们拓扑地塌缩为点，那么我们得到一个新的拓扑球面。 PI将研究商映射何时可以是共形的，并将从这个一般的角度考虑一些具体问题，包括共形焊接，John域的特征，Koebe猜想，Kleinian群的构造和其他动态对象。 PI将继续他的早期工作的几何Kleinian极限集和行为的尺寸，因为我们变形的极限集。 PI还将继续他的工作之间的联系计算几何，双曲几何和共形映射，并寻求新的算法，计算黎曼映射快速和严格的误差估计。 特别是，他将研究计算共形映射使用中轴（一个对象从计算几何），这是密切相关的三维双曲几何通过convexhulls.Conformal映射是重要的，因为他们的中心作用在众多的数学问题（复杂的分析，dynamicalsystems，.）并且在各种应用中（流体流动、脑映射、统计物理、微分方程的数值分析等），所以我们必须对这些地图有一个很好的理论理解和在实践中计算它们的好方法。该建议通过研究与数学和计算机科学的其他部分（点集拓扑，三维双曲几何，Voronoi图）的新联系，加深了我们对共形映射的理论理解，并寻求利用这些联系来发明计算共形映射的新算法。 例如，中轴是计算机科学中广泛研究的对象（它是对物体形状的描述，在模式识别，机器人运动，生物学等方面有许多应用）。PI发现它也可以用来粗略但快速地近似共形映射。PI将寻求改进这种方法，给出更好的共形映射算法，并更好地理解中轴，这将影响其其他应用（例如，当被描述的对象变化很小时，中轴线可能会急剧变化;这是一个严重的计算问题，可以通过将中轴视为双曲几何而不是通常的欧几里得几何中的对象来解决。中轴和共形映射之间的联系也可能为三维空间的研究应用带来新的思路（其中共形映射不存在，但基于中轴的映射仍然存在）。三维问题是应用中最重要的问题，但我们对二维问题的认识远远落后于二维问题，因此，推广到高维的二维问题的新思想是重要的。