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.
