Analysis of Topological Singularities and Their Dynamics

林方华，纽约大学新闻研究所。摘要：本文第一部分研究了Landau-Lefschitz方程中的磁涡（薛定谔和波图与磁场耦合）、被困稀玻色-爱因斯坦凝聚体中的涡旋和涡环（某些非线性薛定谔系统与势耦合）等几种具体情况下拓扑奇点的动力学和稳定性。第二部分是关于Sobolev映射的拓扑奇异性。我们将研究奇点对光滑映射密度的影响，映射空间的拓扑结构，各种能量最小化的规则性和奇点。特别感兴趣的是奇异集的整体结构和能量极小值在这种拓扑奇异点附近的渐近行为。许多有趣的自然现象都包含某种奇特的行为，它们往往通过能量的集中表现出来。因此，描述这些现象的偏微分方程解的奇点是facet的重要组成部分。我们在此提议建立一个严谨的数学理论，可能是关于这类奇点中最重要的一类，拓扑奇点。拓扑奇点出现在许多物理问题中，多年来一直是一个重要的研究课题。已知的例子有铁磁（或反铁磁）连续体中的磁泡，玻色-爱因斯坦凝聚体和超导中的涡旋，液晶中的拓扑缺陷，以及高能物理一般模型中的粒子样解——天子、单极子和瞬子。这些奇异点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，它们通常在能量和动态上更容易观察和稳定。因此，为这种奇点建立一个严谨的数学理论不仅在数学上具有挑战性和趣味性，而且在科学上也具有重要的基础意义。

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.