FRG: Collaborative Research in Gauge Theory

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Understanding of the structure of the four-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. Gauge theory is a crucial tool for the study of the mathematical structures that provide the context for standard models of the physical world. The research supported by this project will develop new mathematical tools and theories that will help test such models and advance our understanding of four-dimensional spaces. A key mathematical idea is the interaction between three-dimensional theories known as Floer theories and the study of invariants of four-dimensional spaces. In addition the project provides research training opportunities for graduate students.This project will develop and extend invariants including abelian gauge theoretic models, cobordism invariants arising from the critical values of Morse like functions such as Daemi's Gamma invariant, equivariant extensions such as equivariant singular instanton knot homology, and new parameterized invariants that combine Konno adjunction-style complexes with homotopy invariants of families. It will also translate structures uncovered in some gauge theory packages to other packages, for example translating L-space notions from Heegaard Floer theory to analogues in the instanton case, or equivariant structures from the Seiberg-Witten Floer theory to the Heegaard Floer theory. Topological applications will include new insights into knot concordance, homology cobordism, diffeomorphism groups of 4-manifolds, and SU(2) representations of 3-manifold groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.