RUI: Manifolds with Density and Isoperimetric Problems

弗兰克·摩根和他的学生将研究密度流形，这是黎曼流形的一种推广，长期以来在概率方面表现突出，对几何及其应用的兴趣迅速增长。密度函数对体积和面积的权重相等，不像公制的保角变化。具有密度的流形是Gromov的mm空间的光滑情况，尽管我们也考虑奇点。宏伟的目标是将黎曼几何的适当部分推广到具有密度的流形。进步应该提高我们对黎曼几何的理解；例如，里奇曲率对有密度的流形有很多不同的推广，而这恰好与黎曼流形相吻合。等周问题提供了一个很好的切入点。高斯空间上的等周定理是密度流形的首要例子，在概率论、黎曼几何的等周问题，特别是佩雷尔曼关于庞加莱猜想的工作中都有应用。方法将包括几何测量理论、黎曼几何和二次变分的标准和创新应用。具有奇点的空间是理论和应用的核心，而等周问题再次提供了一个极好的切入点。我们在一维流形（比如二维曲面或三维宇宙，可能有奇点）上考虑的密度与新生物理中考虑的密度相同，权重随点而变化。这种密度在数学中自然出现；最近的应用包括物理学中的布朗运动，股票期权定价，以及佩雷尔曼关于庞加莱猜想的工作。研究这种自然推广为经典几何提供了新的见解。作为一个重要的例子，对于单位密度的经典几何，自古希腊时代以来一直是几何研究中心的“等周问题”寻求一个给定体积的周长最小的区域；在欧几里得空间中，解是一个圆球。在变密度流形上，等周问题寻求给定加权体积的最小加权周长区域。在更一般的环境下的工作提供了在经典环境下的新结果。摩根的本科生已经解决了一些有趣的案例，并在继续这项工作。摩根广泛地在各种场合演讲，从流行论坛、高中、面向学生和年轻教师的暑期学校到大学座谈会和研究研讨会。

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.