Mathematical Sciences: RUI: Geometry

弗兰克·摩根将继续他的行为研究， m维面积最小化曲面的结构。肥皂膜 是这种表面的典型例子。目前的许多工作 是为了研究它们的奇点。 奇点出现在没有明确定义的 切平面Almgren的工作建立了 奇点集的大小，但到目前为止还知之甚少 关于他们的结构。特别是，他们甚至可能会变成 有分数维。摩根将涉及本科生 学生在这些调查，从而使他们暴露在 现代数学研究的兴奋。他们将参与 在各种计算和实验方面的工作。 摩根将加入科林亚当斯的工作是一个更 拓扑性质他将研究双曲方程的尖点体积 三维流形 摩根所采用的技术属于几何 测度理论这是一个非常强大的理论， 只有相对较少的数学家才能接触到。摩根将使用 这些方法研究了m的奇异集的结构， 最大限度地减少电流的尺寸区域。此外，他还将利用 校准以找到面积最小化的更多示例 表面。这些是封闭的微分形式， 这些曲面的切线空间上的值。他们最近 的子流形的研究中起了很大的作用。 格拉斯曼人

Frank Morgan will continue his studies of the behavior and structure of m-dimensional area minimizing surfaces. Soap films are the prototypical examples of such surfaces. Much current work is directed towards the study of their singularities. Singularities occur at points where there is no well defined tangent plane. Work of Almgren has established upper bounds on the sizes of the singularity sets, but so far little is known about their structure. In particular they may even turn out to have fractional dimension. Morgan will involve undergraduate students in these investigations and thereby expose them to the excitement of modern mathematical research. They will be involved in various computational and experimental aspects of the work. Morgan will be joined by Colin Adams whose work is of a more topological nature. He will study the cusp volumes of hyperbolic three dimensional manifolds. The techniques to be employed by Morgan belong to geometric measure theory. This is an extremely powerful theory which is accessible to relatively few mathematicians. Morgan will use these methods to study the structure of singularity sets of m dimensional area minimizing currents. In addition he will use calibrations to find further examples of area minimizing surfaces. These are closed differential forms which take extreme values on the tangent spaces of such surfaces. They have recently been used to great effect in the study of submanifolds of Grassmannians.