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

9625392 Smyth这个项目属于微分几何和子流形理论领域。研究人员计划继续研究Caratheodory猜想(三维空间中的球面必须至少有两个脐线)，两个变量的连续可微映射的稳定性猜想，以及极小曲面的函数论方面。此外，他们还将研究超曲面的主曲率。子流形理论研究欧几里得空间中的曲线和(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。这个项目包含了这一领域的两个杰出的猜想；试图与其他数学领域，如动力系统理论和代数几何相结合。

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.