Analysis of Topological Singularities and Their Dynamics

林芳华，纽约大学柯朗研究所。DMS-0201443摘要： 这个建议的第一部分是研究几种具体情况下拓扑奇点的动力学和稳定性，包括：Landau-Lefschitz方程中的磁涡旋（薛定谔和波图与磁场耦合）、涡旋和涡环（某些与位势耦合的非线性Schrodinger系统）.第二部分是Sobolev映射的拓扑奇点.我们将研究奇点对光滑映射的稠密性、映射空间的拓扑、各种能量极小的正则性和奇异性，特别是奇异集的全局结构和能量极小在拓扑奇异点附近的渐近行为。 许多有趣的自然现象都含有某种奇异行为，它们常常通过能量集中来表现。因此，描述这些现象的偏微分方程解的奇异性是方面的一个重要组成部分。本文提出建立一个严格的数学理论，它可能涉及一类最重要的奇异性。拓扑奇点。拓扑奇点出现在许多物理问题中，并且在过去的许多年里一直是许多研究的重要课题。在已知的例子中，有铁磁中的磁泡。（或反铁磁）连续体，玻色-爱因斯坦凝聚体中的涡旋和超导性，液晶中的拓扑缺陷，以及skyrminor，单极子和瞬子是高能物理一般模型中的类粒子解。这些奇异性不仅携带着确定的拓扑信息，而且还携带着量化的能量。正因为如此，它们在能量上往往更容易观测和稳定，因此，一个严格的数学理论，这样的奇点将不仅是具有挑战性和有趣的，而且在科学中具有根本的重要性。

Fang-Hua Lin, NYU Courant Institute. DMS-0201443Abstract: The first part of this proposal is to study the dynamics and thestability of topological singularities in several concrete cases including:magnetic vortices in the Landau-Lefschitz equations(Schrodinger and wavemaps coupled with magnetic fields),vortices and vortex rings in the trappeddilute Bose-Einstein condensates(certain nonlinear Schrodinger systemscoupled with potentials).The second part of the proposal is concerned withtopological singularities of Sobolev mappings.We shall examine the effectsof singularities on the denisity of smooth maps,the topology of mappingspaces,the regularity and singularity of various energy minimizers.Ofparticular interest is the global structures of singular sets and asymptotic behavior of energy minimizers near such topological singularities. Many interesting natural phenomena contain some sort singular behaviorand they are often manifiested through energy concentrations.Singularitiesof solutions of partial differential equations which describe thesephenomena are,therefore,an important part of facets.We propose here toestablish a rigourous mathematical theory concerning probabilly a mostimportant class of such singularities ,topological singularity.Topologicalsingularities arise in many physical problems and have been an importantsubject of much study over past many years.Among known examples aremagnetic bubbles in a fereomagnetic(or anti-ferromagnetic)continuum,vortices in Bose-Einstein condensates and superconductivity,topological defects inliquid crystals,as well as skyrminors,monopoles and instantons which areparticle like solutions in generic models of high energy physics.Thesesingularities not only carry definite topological informations but alsoquantified amount of energies.Because of this they are often moreobservable and stable energitically and dynamically.Thus a rigourousmathematical theory for such singularity would be not only mathematicallychallenging and interesting but also of fundamental importance in sciences.