Isoperimetric Problems and Singular Geometry

项目编号：dms -9876471首席研究员：Frank morgan首席研究员将研究欧几里得空间和更一般的黎曼流形中的各种等周问题。他将研究Aubin的猜想，即欧几里得等周不等式在非正曲率的单连通空间中继续成立，至少在小体积中是这样。欧几里得空间中更一般的等周问题涉及几个给定体积的团簇，晶体能量依赖于方向，以及不混相流体，其界面成本取决于分离的流体。他将研究一般体积和高维空间的双泡猜想。许多这类问题的一个显著特征是奇点的出现。等周问题考虑包围体积的代价：肥皂泡包围空气体积，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。奇点通常起着至关重要的作用：肥皂膜三次相遇，材料的结构缺陷，宇宙中的黑洞。基本问题仍然悬而未决。例如，双气泡猜想认为，我们所熟悉的双层肥皂泡是包围和分离两个给定体积空气的最小面积方法。Hass和Schlafly最近对等体积情况的计算机证明可以追溯到Morgan在NSF的本科生研究几何小组的工作，该小组将继续研究一般问题。摩根一年大约做四十次演讲。他喜欢向大众解释，要了解宇宙的几何形状，首先要了解肥皂泡的几何形状。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，目前正在为出版广受欢迎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.