Elliptic problems in field theories

摘要奖：DMS-9972300首席研究员：杨益松人们普遍认为，基本相互作用是由类孤子结构描述的，类孤子结构是各种场论模型的经典解。在拟议的研究中，PI的主要目标是理解场论中四种密切相关的重要孤子类型：电弱双子、共存涡旋和反涡旋、Faddeev类结解和Born-Infeld解。电弱双子是Weinberg-Salam统一理论中关于电磁和弱力的局域解，存在问题是一个含有不定作用量泛函的变分问题，涉及微妙的约束不变量和微妙的弱紧致性问题。共存的涡旋和反涡旋出现在一种新发现的场论中，PI建议开展全面的存在分析，并研究解的拓扑学和解析结构。类稳定结结构在描述包括粒子相互作用和DNA链干扰在内的几个基本过程中具有重要意义。PI计划研究其数学存在理论和动力学性质。由于Born-Infield理论与超弦和超对称的某些领域有关，最近受到了极大的关注。PI将研究源于Born-Infeld理论的经典解及其与几何和物理的联系。这些数学问题不仅有趣和具有挑战性，而且在超导和超流体、粒子物理、宇宙学和分子生物学等广泛领域都有重要的应用。在教育方面，PI已经参与了基于数学应用的本科课程模块的开发。这项提案中提出的研究将为展示数学和其他自然科学领域之间的生动互动提供新的想法。

AbstractAward: DMS-9972300Principal Investigator: Yisong YangIt is widely recognized that fundamental interactions aredescribed by soliton-like structures which are classicalsolutions of various field-theoretical models. The PI's mainobjective in the proposed research is to understand four closelyrelated important types of solitons in field theories:electroweak dyons, coexisting vortices and antivortices,Faddeev's knot-like solutions, and the Born--Infeld solutions.Electroweak dyons are electrically and magnetically chargedlocalized solutions in the Weinberg--Salam unified theory forelectromagnetic and weak forces in particle physics and theexistence problem is a variational problem with indefinite actionfunctional involving delicate constraint invariants and subtleweak compactness issues. The coexisting vortices andantivortices appear in a newly discovered field theory and the PIproposes to develop a full existence analysis and study thetopological and analytic structures of the solutions. Stableknot-like structures are important in describing severalfundamental processes including particle interactions and DNAchain interferences. The PI plans to study its mathematicalexistence theory and dynamical properties. The Born--Infeldtheory has received much attention recently due to its relevancein some areas of superstrings and supersymmetry. The PI willstudy classical solutions arising from the Born--Infeld theoryand their connections with geometry and physics.These mathematical problems are not only interesting andchallenging, but also have important applications in a wide rangeof areas including superconductivity and superfluids, particlephysics, cosmology, and molecular biology. On the educationalside, the PI has already been involved in developingundergraduate curriculum modules based upon applications ofmathematics. The research presented in this proposal will providenew ideas for demonstrating the vivid interactions betweenmathematics and other areas of natural science.