RUI: Manifolds with Density and Isoperimetric Problems

RUI:具有密度和等周问题的流形

基本信息

  • 批准号:
    0803168
  • 负责人:
  • 金额:
    $ 14.54万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2008
  • 资助国家:
    美国
  • 起止时间:
    2008-09-01 至 2012-08-31
  • 项目状态:
    已结题

项目摘要

Frank Morgan and his students will study manifolds with density, a generalization of Riemannian manifolds, long prominent in probability and of rapidly growing interest in geometry and applications. The density function weights volume and area equally, unlike a conformal change of metric. Manifolds with density are the smooth case of Gromov's mm spaces, although we also consider singularities. The grand goal is to generalize appropriate parts of Riemannian geometry to manifolds with density. Advances should improve our understanding of Riemannian geometry; for example, Ricci curvature has many different generalizations to manifolds with density, which just happen to coincide for Riemannian manifolds. Isoperimetric problems provide an excellent entry point. Isoperimetric theorems on Gauss space, the premier example of a manifold with density, have had applications in probability theory, in isoperimetric problems in Riemannian geometry, and specifically in Perelman's work on the Poincaré Conjecture. Methods will include standard and innovative applications of geometric measure theory, Riemannian geometry, and second variation. Spaces with singularities are central to theory and applications, and isoperimetric problems once again provide an excellent entry point. The density we consider on a ann-dimensional manifold (such as a 2-dimensional surface or a 3-dimensional universe, perhaps with singularities), is the same kind of density one considers in freshman physics, a weighting that varies from point to point. Such densities arise naturally throughout mathematics; recent applications include Brownian motion in physics, stock option pricing, and Perelman's work on the Poincaré Conjecture. Studying this natural generalization provides new insights into classical geometry. As one important example, for the classical geometry of unit density, the _isoperimetric problem_, central to geometry research since the time of the Ancient Greeks, seeks a region of given volume of least perimeter; in Euclidean space, the solution is a round ball. On a manifold with variable density, the isoperimetric problem seeks the region of given weighted volume of least weighted perimeter. Work in the more general context provides new results in the classical context. Morgan's undergraduate research students have solved interesting sample cases and are continuing the work. Morgan lectures widely at venues ranging from popular forums, high schools, and summer schools for students and young faculty to university colloquia and research seminars.
弗兰克·摩根和他的学生将研究具有密度的流形,这是黎曼流形的推广,它在概率论中长期突出,并且对几何和应用的兴趣迅速增长。密度函数对体积和面积的权重是相等的,与密度的共形变化不同。是格罗莫夫毫米空间的光滑情况,尽管我们也考虑奇点,但我们的宏伟目标是将黎曼几何的适当部分推广到具有密度的流形。黎曼几何;例如,里奇曲率对密度流形有许多不同的推广,这恰好与黎曼流形问题相一致,为高斯空间(密度流形的首要例子)提供了一个很好的切入点。在概率论、黎曼几何中的等周问题以及特别是佩雷尔曼关于庞加莱方程的工作中都有应用猜想。方法将包括几何测度理论、黎曼几何和二阶变分的标准和创新应用,奇点空间是理论和应用的核心,等周问题再次提供了一个很好的切入点。维度流形(例如二维表面或三维宇宙,可能具有奇点)与新生物理学中考虑的密度相同,这种密度的权重因点而异。自然地贯穿整个数学;最近的应用包括物理学中的布朗运动、股票期权定价和佩雷尔曼对庞加莱猜想的研究,为经典几何提供了新的见解,例如单位密度的经典几何。自古希腊时代以来,问题就是几何研究的核心,它寻求给定体积的最小周长的区域;在欧几里得空间中,解决方案是在密度可变的圆球上。等周问题寻求给定加权体积的最小加权周长的区域。摩根的本科生研究人员已经解决了有趣的示例案例,并且正在广泛地继续进行这项工作。为学生和年轻教师举办的热门论坛、高中和暑期学校以及大学座谈会和研究研讨会。

项目成果

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Frank Morgan其他文献

Protein translation components are colocalized in granules in oligodendrocytes.
蛋白质翻译成分共定位于少突胶质细胞的颗粒中。
  • DOI:
    10.1016/j.memsci.2022.120786
  • 发表时间:
    1995-08-01
  • 期刊:
  • 影响因子:
    4
  • 作者:
    E. Barbarese;Dennis E. Koppel;M. P. Deutscher;C;ra L. Smith;ra;K. Ainger;Frank Morgan;J. Carson
  • 通讯作者:
    J. Carson
Psychometric assessment of the Edinburgh Postnatal Depression Scale in an obstetric population
爱丁堡产后抑郁量表对产科人群的心理测量评估
  • DOI:
    10.1016/j.psychres.2020.113161
  • 发表时间:
    2020-06-01
  • 期刊:
  • 影响因子:
    11.3
  • 作者:
    Molly M. Long;R. Cramer;L. Bennington;Frank Morgan;Charles A. Wilkes;Arlene J. Fontanares;Nikki Sadr;S. Bertolino;J. Paulson
  • 通讯作者:
    J. Paulson
CHAPS-D: The Compact Hyperspectral Air Pollution Sensor–Demonstrator
CHAPS-D:紧凑型高光谱空气污染传感器 - 演示器
  • DOI:
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Benjamin Stewart;William H. Swartz;Frank Morgan;Walter Zimbeck;Trevor Palmer;Joseph Linden;Ryan Newport;Jake Strang;Gerard Otter;Floris van Kempen;Sanne Van De Boom;Ivan Ferrario
  • 通讯作者:
    Ivan Ferrario
Training Effective Interpreters for Diabetes Care and Education
培训有效的糖尿病护理和教育口译员
  • DOI:
  • 发表时间:
    2006
  • 期刊:
  • 影响因子:
    0
  • 作者:
    M. McCabe;D. Gohdes;Frank Morgan;Joanne Eakin;Cheryl Schmitt
  • 通讯作者:
    Cheryl Schmitt
The informed consent process in a cross-cultural setting: is the process achieving the intended result?
跨文化环境中的知情同意过程:该过程是否达到了预期结果?
  • DOI:
    10.1517/14728214.13.3.393
  • 发表时间:
    2005
  • 期刊:
  • 影响因子:
    3.2
  • 作者:
    M. McCabe;Frank Morgan;Helen Curley;Richard M. Begay;D. Gohdes
  • 通讯作者:
    D. Gohdes

Frank Morgan的其他文献

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{{ truncateString('Frank Morgan', 18)}}的其他基金

The Williams SMALL REU Site
威廉姆斯小型 REU 站点
  • 批准号:
    0353634
  • 财政年份:
    2004
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant
Minimal Surfaces and Singular Geometry
最小曲面和奇异几何
  • 批准号:
    0203434
  • 财政年份:
    2002
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Standard Grant
Isoperimetric Problems and Singular Geometry
等周问题和奇异几何
  • 批准号:
    9876471
  • 财政年份:
    1999
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Standard Grant
Mathematical Sciences: RUI: Minimal Surfaces, Clusters, and Singular Geometry
数学科学:RUI:最小曲面、簇和奇异几何
  • 批准号:
    9625641
  • 财政年份:
    1996
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: RUI: Geometry, Topology, and Number Theory at Williams
数学科学:RUI:威廉姆斯的几何、拓扑和数论
  • 批准号:
    9302843
  • 财政年份:
    1993
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: RUI: Geometric Measure Theory and the Topology of 3-Manifolds
数学科学:RUI:几何测度论和3-流形拓扑
  • 批准号:
    9000937
  • 财政年份:
    1990
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Small Geometry Project
数学科学:小几何项目
  • 批准号:
    8900348
  • 财政年份:
    1989
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: RUI: Geometry
数学科学:RUI:几何
  • 批准号:
    8802266
  • 财政年份:
    1988
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant

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会议:圣路易斯拓扑会议:3 流形中的流动和叶理
  • 批准号:
    2350309
  • 财政年份:
    2024
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Geodesic arcs and surfaces for hyperbolic knots and 3-manifolds
双曲结和 3 流形的测地线弧和曲面
  • 批准号:
    DP240102350
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    2024
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    $ 14.54万
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    Discovery Projects
Diffusions and jump processes on groups and manifolds
群和流形上的扩散和跳跃过程
  • 批准号:
    2343868
  • 财政年份:
    2024
  • 资助金额:
    $ 14.54万
  • 项目类别:
    Continuing Grant
CAREER: Nonlinear Finite Element Manifolds for Improved Simulation of Shock-Dominated Turbulent Flows
职业:用于改进冲击主导的湍流模拟的非线性有限元流形
  • 批准号:
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CAREER: Algebraic, Analytic, and Dynamical Properties of Group Actions on 1-Manifolds and Related Spaces
职业:1-流形和相关空间上群作用的代数、解析和动力学性质
  • 批准号:
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