Mathematical Sciences: RUI: Geometric Measure Theory and the Topology of 3-Manifolds

一名研究者将使用校准研究最小面积曲面的行为和奇异结构。在黎曼流形上，校准提供了同调类的杰出代表。从等周不等式、材料科学和肥皂膜的研究中产生的广泛的开放性问题迫使人们考虑比面积更一般的积分和新的表面类别。第二个研究者将继续他的双曲3-流形和3-轨道的体积研究。其中包括球填料的研究，它对双曲型3流形的几何有了新的认识。环面标定的一个例子是一组纵向圆。在这种情况下，校准的元素是同源类生成器。它们的长度最小。一位研究者将使用新兴的校准理论来理解面积最小化曲面上的各种积分。第二名研究员将研究三维曲面或流形的几何形状。

One investigator will study the behavior and singular structure of area-minimizing surfaces using calibrations. On Riemannian manifolds calibrations provide distinguished representatives of homology classes. A wide range of open problems arising from the study of isoperimetric inequalities, materials science, and soap films compel the consideration of integrands more general than area and of new classes of surfaces. A second investigator will continue his studies of volumes of hyperbolic 3-manifolds and 3-orbifolds. These include the study of horoball packings which has shed new light on the geometry of hyperbolic 3-manifolds. An example of a calibration of the torus is the set of longitudinal circles. As in this case, elements of a calibration are homology class generators. These have minimal length. One investigator will use the emerging theory of calibrations to understand assorted integrals on area-minimizing surfaces. The second investigator will study the geometry of three-dimensional surfaces, or manifolds.