Mathematical Sciences: Differential Geometry and Geometric Analysis

调查人员将继续之前在微分几何和几何分析方面的一些项目的工作。杰夫·切格计划更深入地了解与某些椭圆算子族有关的不变量，这些算子族最近在理论物理中引起了一些兴趣。他还提议进行广泛的工作，以进一步深入了解完备黎曼流形的有界几何、折叠和特征数之间的相互作用。此外，他还将继续用分段线性流形来扩展他的光滑流形曲率的逼近理论。Gromoll期望在他为具有非负Ricci曲率的完备黎曼空间建立全局结构理论方面取得进一步的实质性进展，特别是有限结果和额外的例子类的构造。他还想继续分析度量叶的刚性现象，即非负曲线流形中的局部处处等距划分，这已被证明是此类空间整体几何的一个关键而微妙的方面。在子流形理论领域，他将致力于最近关于欧几里德子流形的等距变形的全局结果的各种扩展，以及探索极小实Kaehler子流形的新理论，极小实Kaehler子流形是在此背景下出现的一类特殊但特别有趣的极小子流形。埃宾计划在三个领域开展工作，这三个领域都涉及自由边值问题。首先是对薄膜运动的研究。第二个是液滴动力学的研究，第三个是大原子核的液滴模型。第二和第三个区域都被作为约束到函数空间的子流形的动力系统的例子来研究。这一分析的关键一步是研究它们的相对曲率。Teleman打算继续他关于Novikov猜想的工作，Novikov猜想是微分拓扑学和算子理论中的一个突出问题。他还希望发现一种新的方法来研究Pontrjagin类的组合流形，这在理论物理学中也会有很大的兴趣。

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.