Instantons, Representations and Low Dimensional Topology

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Low dimensional topology is an area of mathematics that studies qualities of three- and four-dimensional spaces which are insensitive to continuous deformations such as stretching and bending. These spaces model real world objects and low dimensional topology is highly relevant to other scientific disciplines. For example, knot theory, a branch of low dimensional topology, is an effective tool in studying configurations of protein and DNA. In addition, topology plays an essential role in formulating modern theories in physics. Perhaps more surprisingly, tools from modern physics, more specifically quantum filed theory, have yielded significant progresses in low dimensional topology. This National Science Foundation funded project promotes systematic application of ideas in physics to topology and vice versa. The PI aims to investigate applications of the Yang-Mills theory of high energy physics in the topological properties of three-and four-dimensional objects. The proposed research also partly focuses on foundational questions in symplectic geometry, a field with close ties with Physics.Instanton Floer homology, defined using Yang-Mills gauge theory, provides algebraic invariants of three- and four-dimensional manifolds. The PI will apply different versions of instanton Floer homology to the study of problems in low dimensional topology. The focus of the first part of the project is the Atiyah-Floer conjecture. This conjecture states that one can apply methods from symplectic geometry to define three-manifold invariants. Furthermore, the resulting invariant, often called symplectic instanton Floer homology, is isomorphic to instanton Floer homology. The PI and his collaborators will develop tools in symplectic topology which can be used to construct new types of symplectic instanton Floer homology. They will also use a certain partial differential equation, called the mixed equation, to address various versions of the Atiyah-Floer conjecture. Another goal of this project is to prove the existence of non-trivial representations of knot groups into the special unitary group SU(N). An outcome of this project would be to give gauge theoretical proofs of the Smith conjecture and the Covering Conjecture. The final goal of this project is to employ instanton Floer homology in the study of the homology cobordism group. The PI recently constructed a family of new invariants for the homology cobordism group using Yang-Mills gauge theory. The PI will apply these invariants to better understand the structure of the homology cobordism group.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.