Research in Koebe Uniformization and Circle Packings

小行星9622068 这个项目在于复杂的分析，拓扑和几何的接口。它涉及共形几何和圆域在平面上。相对圆域的共形映射推广了由同一组合模式的圆填充导出的映射。研究人员将试图确定是否经典的结果，圈包装地图持有这些广义映射以及。此外，研究者将研究拓扑流体力学中的以下问题：估计在某些理想物理条件下受到变形的不可压缩向量场的能量下限。 圆填充和圆映射问题是涉及数学几个不同分支的经典问题。例如，最近发现，涉及负曲率的三维双曲几何在圆图案和填充问题中起着重要作用。此外，某些极值包装问题及其高维类似物-其解决方案可以近似与计算机模拟-可能有应用在工业包装问题。

9622068 He This project lies in the interface of complex analysis, topology and geometry. It deals with conformal geometry and circle domains in the plane. Conformal mappings of relative circle domains generalize mappings induced from circle packings of the same combinatorial pattern. The investigator will try to determine whether classical results for circle packing maps hold for these generalized mappings as well. In addition, the investigator will study the following problem in topological fluid mechanics: estimate the lower bound for the energy of incompressible vector fields which are subject to deformations under some ideal physical condition. The circle packing and circle mapping problems are classical problems touching on several different branches of mathematics. For example, recently it was discovered that three-dimensional hyperbolic geometry involving negative curvature plays an important role in circle pattern and packing problems. Also, certain extremal packing problems and their higher dimensional analogs - whose solutions can be approximated with computer simulation - may have applications in industrial packing problems.