Isoperimetric Problems and Singular Geometry

摘要奖：DMS-9876471主要研究人员：弗兰克·摩根主要研究欧氏空间和更一般的黎曼流形中的各种等周问题。他将致力于Aubin的猜想，即欧几里得各向异性等周不等式继续在具有非正曲率的简单连通空间中成立，至少对小体积是这样。欧氏空间中更一般的等周问题涉及几个给定体积的集合体、与方向有关的结晶能量以及不相容流体，对于不相容流体，界面成本取决于所分离的流体。他将研究一般体积和更高维度的双泡沫猜想。其中许多问题的一个显著特征是出现奇点。等周问题考虑了封闭体积的成本：包围空气体积的肥皂泡，材料中分子的结晶排列，宇宙曲率中面积和体积之间的相互作用。奇点通常扮演着至关重要的角色：肥皂片三人相遇，材料的结构缺陷，宇宙中的黑洞。根本性的问题仍然悬而未决。例如，双泡猜想说，熟悉的双皂泡泡是封闭和分离两个给定体积的空气的最小面积的方法。Hass和Schlafly最近对等体积情况的计算机证明可以追溯到摩根的NSF本科生研究几何小组的工作，该小组将继续研究这个一般问题。摩根每年举行约40次会谈。对于通俗的听众，他喜欢解释，理解宇宙几何的方法首先是理解肥皂泡的几何。他有一个数学聊天电视节目和专栏，都可以在美国数学协会的网站http://www.maa.org，上看到，目前正在为出版流行的数学聊天书做最后的准备。

AbstractAward: DMS-9876471Principal Investigator: Frank MorganThe principal investigator will study various isoperimetricproblems in Euclidean space and more general Riemannianmanifolds. He will work on Aubin's conjecture, that the Euclideanisoperimetric inequality continues to hold in a simply connectedspace of nonpositive curvature, at least for small volumes. Moregeneral isoperimetric problems in Euclidean space involveclusters of several given volumes, crystalline energies dependenton direction, and immiscible fluids, for which interface costdepends on the fluids separated. He will work on the DoubleBubble Conjecture for general volumes and in higher dimensions. Adistinguishing feature of many of these problems is theappearance of singularities.Isoperimetric problems consider the cost of enclosing volume:soap bubbles enclosing volumes of air, crystalline arrangementsof molecules in materials, the interplay between area and volumein the curvature of the universe. Singularities often play acritical role: soap films meeting in threes, structural defectsin materials, black holes in the universe. Fundamental questionsremain open. The Double Bubble Conjecture, for example, says thatthe familiar double soap bubble is the least-area way to encloseand separate two given volumes of air. The recent computer prooffor the case of equal volumes by Hass and Schlafly can be tracedback to work by Morgan's NSF undergraduate research GeometryGroup, which will continue to work on the general problem. Morgangives some forty talks a year. To popular audiences he likes toexplain that the way to understand the geometry of the universeis first to understand the geometry of soap bubbles. He has aMath Chat TV show and column, both available at the MathematicalAssociation of America website at http://www.maa.org, and iscurrently making final preparations for the publication of apopular Math Chat Book.