Mathematical Sciences: RUI: Geometry

弗兰克·摩根将继续研究m维面积最小化曲面的行为和结构。肥皂膜就是这种表面的典型例子。目前的许多工作都是针对它们的奇点进行研究。奇点出现在没有明确定义的切平面的点上。Almgren的工作已经建立了奇点集大小的上限，但到目前为止，人们对奇点集的结构知之甚少。特别是，它们甚至可能被证明是分数维的。摩根将让本科生参与这些调查，从而使他们接触到现代数学研究的兴奋。他们将参与这项工作的各种计算和实验方面。科林·亚当斯将加入摩根的行列，他的工作更具拓扑性。他将研究双曲三维流形的尖点体积。摩根所采用的技术属于几何测度论。这是一个极其强大的理论，只有相对较少的数学家能够理解。摩根将使用这些方法来研究m维面积最小化电流的奇点集的结构。此外，他还将使用校准来寻找更多面积最小化曲面的例子。这些是封闭的微分形式，它们在这类曲面的切线空间上取极值。最近，它们在Grassmannia子流形的研究中取得了很大的效果。

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.