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

建议编号：DMS-0103888韩的建议包括两个研究项目和一个教育项目。在第一个研究项目中，韩建议采用几何分析的最新技术来研究二维欧拉方程中涡片演化的某些几何和分析问题。特别是，该项目提出了一种可能的可纠变变折的弱解概念，如果成功，它将比目前可用的基于可积涡度函数的弱解概念对奇异涡片的演化提供更多的几何描述。在一些原型情况下，韩还建议研究相交涡片卷曲中更精确的几何行为。在第二个项目中，han提出了负曲面调和映射的一个值分布理论.最近这一领域的结果表明，这类调和映射在无穷远附近的诱导分层结构和树状结构中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.几何分析和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，韩建议重振主要面向数学教育专业的本科几何课程和本科微分几何课程，以更好地满足更广泛的受众的需求，这些受众是在当今快速变化的技术环境和新的交叉学科领域的出现而成长起来的。很好地理解集中涡旋的演变，如涡片，具有非常重要的实用价值。控制涡面演化的数学方程具有许多与近年来几何分析中成功研究的特征非常相似的特征。PI希望，来自各个领域的想法和工具的相互作用将在更好地理解涡片演化的几何方面带来一些丰硕的结果。在他的第二个项目中，PI还希望将几何方法和解析方法更紧密地结合在一起，这种方法的成功可能会让我们进一步深入了解纯粹的几何或分析问题。第三个拟议项目的成功实施将通过更好地培养数学教育专业的学生，为K-12数学教育做出积极贡献。

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.