Geometric Analysis of Vortex Sheet Evolution and Value Distribution of Harmonic Maps into Hadamard Surfaces

提案号：dms -0103888韩的提案包括两个研究项目和一个教育项目。在第一个研究项目中，他提出采用最新的几何分析技术来研究二维欧拉方程中涡片演化中的某些几何和分析问题。特别是，该项目提出了一个可能的可整流变分弱解的概念，如果成功，将比目前可用的可积涡度函数弱解的概念提供更多的几何描述奇异涡片的演变。在一些原型情况下，Han还建议研究相交涡旋片卷起来时更精确的几何行为。在第二个项目中，Han提出建立一个负曲面调和映射的值分布理论。这一领域最近的研究结果表明，在无穷近处的诱导叶状结构和树状结构中，这种调和映射具有非常丰富的几何行为。这种行为的几何描述与相关偏微分方程组的渐近边值问题的可解性密切相关。几何分析、偏微分方程的相互作用，以及它们在一些有趣的应用分析问题中的应用，是连接这两个项目的主线。在教育项目中，他建议重新开设一门主要面向数学教育专业的本科几何课程，以及一门本科微分几何课程，以更好地满足当今快速变化的技术环境和新的跨学科领域出现的更广泛受众的需求。深入了解旋涡片等集中涡的演化，具有极其重要的实用价值。控制涡片演化的数学方程显示出许多与近年来在几何分析中成功研究的特征非常相似的特征。PI希望来自各个领域的思想和工具的相互作用将在更好地理解涡旋片演化的几何方面带来一些富有成效的结果。在他的第二个项目中，PI还希望将几何和分析方法更紧密地结合起来，其成功可能为纯几何或分析问题提供进一步的见解。如果第三次计划顺利实施，将会对K-12数学教育做出积极的贡献，培养数学教育专业的学生。

Proposal Number: DMS-0103888Han's proposal comprises of two research projects and an education project.In the first research project Han proposes to adapt recent techniques fromgeometric analysis to study certain geometrical and analytical problems inthe evolution of vortex sheets in two dimensional Euler equations. In particular, the project proposes to study a possible notion of weak solution in terms of rectifiable varifolds, which, if successful,should provide more geometric description to the evolution of singular vortex sheet than the currently available notion of weak solution in terms of integrable vorticity functions.In some prototype situations, Han also proposes to study the more precisegeometric behavior in the roll up of intersecting vortex sheets. In the second project, Han proposes to establish a value distribution theory for harmonic maps into negatively curved surfaces.Recent results in this area have suggested very rich geometric behavior of such harmonic maps in terms of induced foliation structuresand tree-like structures in the vicinity of infinity.The geometric description of such behavior is closely related tothe solvability of asymptotic boundary value problems of the relevantsystem of partial differential equations. The interaction of geometric analysis, partial differential equations, and their application to some interesting applied analysis problems is the thread connecting the two projects. In the education project, Han proposes to rejuvenate an undergraduate geometry course mostly for math education majors and an undergraduate differential geometry course to better serve the need of a wider audience that has grown out of today's rapidly changing technological environmentsand emergence of new interdisciplinary fields.A good understanding of the evolution of concentrated vortices, suchas vortex sheet, has immensely important practical values. The mathematical equations that govern the evolution of vortex sheetsexhibit many features very similar to those that have beensuccessfully studied in geometric analysis in recent years. The PI hopesthat the interactions of ideas and tools from across the fields willbring some fruitful results in understanding better the geometric aspectsof vortex sheet evolution. In his second project, the PI also hopes tocombine closer the geometric and the analytic approaches, the success of whichmay provide further insight back to purely geometrical or analytical problems.The successful implementation of the third proposed project will will be a positive contribution to the K-12 math education through better training of math education majors.