Gilmer will continue studying integral topological quantum theories. With Gregor Masbaum, he will study representations of mapping class groups that can be defined using integral topological quantum theories. He also will study a universal representation of the Torelli group that Masbaum has defined. Gilmer plans to compute strong shift equivalence class invariants of knots and other spaces, which can be defined using TQFT. He wishes to calculate further strong shift equivalnce invariants to uncover their topological meaning. He has used topological quantum theory invariants to find obstructions to fibered knots being ribbon knots. He wants to see if there are further obstructions. In general, he will try to use quantum topology as a tool in low-dimensional topology. Together with Charles Livingston, he wishes to understand better which non-orientable surfaces in half 4-space can have boundary a given knot in 3-space. Gilmer also plans to use 4-dimensional topology to study the topology of real algebraic curves in the real projective plane.
Topology is the study of intrinsic shape. It is sometimes called "rubber sheet" geometry as frequently the objects one studies can be twisted and stretched but not torn. Recently topology has experienced a large influx of ideas from physics. Topological quantum field theory is one of the most current and exciting areas of topology with intimate connections to high energy physics, quantum computing as well as other areas of mathematics, for instance number theory. Gilmer will use topological quantum theory as a tool to study low dimensional topology. Low dimensional topology is important for chemistry and biology as it has implications for the mechanism of DNA, and other molecular configurations.
