Mathematical Sciences: Yang-Mills-Higgs Theory on R3 with Positive Coupling Constant

在这个项目中，首席研究员将在一个正耦合常数的情况下研究SU(2) Yang-Mills-Higgs方程。她的目标是证明具有任意正耦合常数的SU(2) Yang-Mills- Higgs泛函在三维空间中的非极小有限作用临界点的存在性，从而推广了Taubes在零耦合常数情况下的工作。Taubes建立了一种适用于规范论中出现的三维变分问题的Liusternik-Schnirelman最小-极大理论的形式。首席研究员希望建立一个类似的结果，适用于正耦合常数的情况。在她的研究过程中，她将与Taubes和其他专家合作。规范论现在是数学物理的基础领域之一。它包含了许多数学家和物理学家感兴趣的深刻而重要的问题。其中一个问题涉及杨-米尔斯理论中出现的三维变分方程的有限作用解的性质。主要研究者将重点研究该领域的一个重要问题，即在耦合常数为正的情况下，Yang-Mills泛函的非极小有限作用临界点的存在性。

In this project the principal investigator will study the SU(2) Yang-Mills-Higgs equations in three-space in the case of a positive coupling constant. Her goal is to show the existence of non-minimal finite-action critical points of the SU(2) Yang-Mills- Higgs functional in three-space with arbitrary positive coupling constant, thereby extending the work of Taubes for the case of zero coupling constant. Taubes was able to establish a form of the Liusternik-Schnirelman min-max theory that applies to the variational problems in three-space that arise in gauge theory. The principal investigator hopes to establish a similar result that applies in the case of positive coupling constant. She will collaborate with Taubes and other experts in the course of her research. Gauge theory is now one of the fundamental areas in mathematical physics. It contains a number of deep and important problems that are of interest to mathematicians and physicists. One of these problems involves the properties of finite-action solutions of certain variational equations that arise in the Yang- Mills theory in three-space. The principal investigator will focus on an important problem in this area, namely showing the existence of non-minimal finite-action critical points of the Yang-Mills functional in the case when the coupling constant is positive.