Elliptic problems in field theories

摘要：dms -9972300项目负责人：杨一松人们普遍认为，基本相互作用是用类孤子结构来描述的，这是各种场理论模型的经典解。在提出的研究中，PI的主要目标是理解场论中四种密切相关的重要类型的孤子：电弱dyons，共存涡旋和反涡旋，Faddeev的结状解和Born- Infeld解。电弱代子是Weinberg- Salam统一理论中电和磁带电的定域解，在粒子物理中具有电磁和弱力，存在性问题是一个具有不定作用泛函的变分问题，涉及微妙的约束不变量和微妙的弱紧性问题。共存涡旋和反涡旋出现在一个新发现的场论中，并提出了一个完整的存在分析和研究解决方案的拓扑结构和解析结构。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是重要的。PI计划研究它的数学存在理论和动力学性质。玻恩-因费尔德理论由于其在超弦和超对称的某些领域的相关性，最近受到了很多关注。PI将研究由玻恩-因菲尔德理论产生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且具有挑战性，而且在包括超导和超流体、粒子物理学、宇宙学和分子生物学在内的广泛领域都有重要的应用。在教育方面，PI已经参与开发基于数学应用的本科课程模块。该提案中的研究将为展示数学与自然科学其他领域之间生动的相互作用提供新的思路。

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.