Isoperimetric Problems and Singular Geometry

AbstractAward：DMS-9876471首席研究员：弗兰克摩根首席研究员将研究各种isoperimetricproblems在欧几里德空间和更一般的黎曼流形。他将致力于奥宾的猜想，即欧几里德等周不等式继续举行在一个简单的connectedspace的非正曲率，至少在小体积。更一般的等周问题在欧几里得空间involveclusters几个给定的体积，结晶能量dependenton方向，和不混溶的流体，其中界面成本取决于分离的流体。他将致力于一般体积和更高维度的DoubleBubble猜想。这些问题的一个显著特征是奇点的出现。等周问题考虑了封闭体积的代价：肥皂泡封闭了大量的空气，材料中分子的结晶体，宇宙曲率中面积和体积之间的相互作用。 奇点经常扮演着关键的角色：肥皂膜三重相交，结构缺陷材料，宇宙中的黑洞。基本问题仍然悬而未决。例如，双气泡猜想说，我们熟悉的双肥皂泡是分离和分离两个给定体积空气的最小面积方法。最近哈斯和施拉夫利对等体积情况的计算机证明可以追溯到摩根的NSF本科研究几何组的工作，该小组将继续研究一般问题。莫甘吉夫一年约有四十次演讲。他喜欢向大众解释，要理解宇宙的几何形状，首先要理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网站http://www.maa.org上找到，目前正在为出版一本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.