TQFT and Low Dimensional Topology

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Gilmer will continue the study of topological quantum field theories defined over rings of cyclotomic integers. He will continue to study with Masbaum the associated mapping class groups representations acting on modules over these rings and also over their finite quotients. He will also study the related infinite dimensional perturbative representation of the Torelli group that Masbaum has defined. Gilmer will study strong shift equivalence class invariants of knots and other spaces which are equipped with an infinite cyclic cover. He will use these structures to study questions in low dimensional topology. Gilmer will also study invariants of links that he has associated to collections of curves in the real projective plane. This should lead to restrictions on topology of real algebraic curves.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 had a large influx of ideas from physics. Topological quantum field theory (TQFT) is an area of topology with intimate connections to high energy physics, quantum computing as well as other areas of mathematics, for instance number theory. The same topological quantum field theories that Gilmer will be exploring have been proposed by others for applications to quantum computing. Gilmer would also like to use TQFT as a tool to study classical questions in topology. Low dimensional topology is important for chemistry and biology as it has implications for the mechanism of DNA, and other molecular configurations.