Mathematical Sciences: Differential Geometry and Geometric Analysis

调查人员将继续以前的工作， 微分几何和几何分析的项目。杰夫 Cheeger计划更深入地了解不变量 与某些椭圆算子族相关联， 最近对理论物理学有点兴趣 他是 还建议开展广泛的工作， 有界几何体的相互作用，塌陷， 完备黎曼流形的特征数 在 此外，他将继续扩大他的近似理论， 光滑流形的分片线性曲率 格罗莫尔 期待着他在努力建立 完备黎曼空间的整体结构理论 非负Ricci曲率，特别是有限性结果， 构建额外的示例类。 他还 他想继续分析 度量叶理，即局部处处等距 分区，在非负弯曲的流形，这已经转向 这是全球几何学中一个关键而微妙的方面， 这样的空间。 在该地区的子流形理论，他将工作 最近关于等距的整体结果的各种推广 变形的欧氏子流形，以及探索新的 极小真实的Kaehler子流形理论，一个特殊但充分的 一类特别有趣的极小子流形， 在这种背景下出现。 Ebin计划在三个领域开展工作， 其中涉及自由边值问题。 首先是 研究薄膜运动的学科。 二是学习 第三个是液滴 一个大的核模型。 第二和第三个领域都是 作为动力系统的例子， 函数空间的子流形 此分析的关键步骤是 研究它们的相对曲率。 Teleman打算 继续他的工作诺维科夫猜想，一个杰出的 微分拓扑和算子理论中的一个问题。 他还 希望发现一种新的方法来Pontrjagin类， 组合流形，这也将是重要的 对理论物理感兴趣。

The investigators will continue previous work on a number of projects in differential geometry and geometric analysis. Jeff Cheeger plans to gain a deeper understanding of invariants associated with certain families of elliptic operators which have recently been of some interest in Theoretical Physics. He is also proposing extensive work toward obtaining further insight into the interaction of bounded geometries, collapsing, and characteristic numbers for complete Riemanian manifolds. In addition, he will continue to expand his approximation theory for curvature of smooth manifolds by piecewise linear ones. Gromoll expects further substantial progress in his efforts to establish a global structure theory for complete Riemannian spaces of nonnegative Ricci curvature, in particular finiteness results and the construction of additional classes of examples. He also wants to proceed with his analysis of rigidity phenomena of metric foliations, i.e. locally everywhere equidistant partitions, in nonnegatively curved manifolds, which has turned out to be a crucial and delicate aspect of the global geometry of such spaces. In the area of submanifold theory he will work on various extensions of recent global results concerning isometric deformation of euclidean submanifolds, as well as explore the new theory of minimal real Kaehler submanifolds, a special but ample class of particularly interesting minimal submanifolds that surfaced in that context. Ebin plans to work in three areas all of which involve free boundary value problems. The first is the study of the motion of a thin membrane. The second is the study of dynamics of a liquid drop, and the third is the liquid drop model for a large nucleus. The second and third areas are both studied as examples of dynamical systems constrained to submanifolds of function spaces. A key step in this analysis is the study of their relative curvature. Teleman intends to continue his work on the Novikov conjecture, an outstanding problem in Differential Topology and Operator Theory. He also wants to discover a new approach to Pontrjagin classes of combinatorial manifolds, which would also be of significant interest in Theoretical Physics.