Mathematical Sciences: Analysis and the Geometry of Submanifolds/Injectivity Problems in Geometry

小行星9625392 这个项目是在微分几何和子流形理论领域。研究人员计划追求Caratheodory猜想（3-空间中的球体必须至少有两个脐点），两个变量中连续可微映射的稳定性猜想，以及极小曲面的函数理论方面。此外，他们将检查超曲面的主曲率。 子流形理论研究欧氏空间中曲线和（超）曲面的各种性质-度量、拓扑、全局和局部。子流形理论中的一个重要考虑是了解子流形的外在性质如何与其内在性质相互作用。这个项目包含两个杰出的acquitures在这一领域;试图接口与其他领域的数学，如动力系统理论和代数几何。

9625392 Smyth This project lies in the area of differential geometry and submanifold theory. The investigator plans to pursue the Caratheodory conjecture (a sphere in 3-space must have at least two umbilics), the stability conjecture for continuously differentiable maps in two variables, and function theoretic aspects of minimal surfaces. In addition, they will examine principal curvatures of hypersurfaces. Submanifold theory deals with various properties - metric, topological, global and local - of curves and (hyper-) surfaces lying in Euclidean and sometimes more general spaces. An important consideration in submanifold theory is to understand how extrinsic properties of a submanifold interact with its intrinsic properties. This project contains two of the outstanding conjectures in this area; attempts to interface with other areas of mathematics such as dynamical systems theory and algebraic geometry.