Hyperkähler manifolds are geometric spaces with special symmetries based on the quaternions, the four-dimensional analogue of the complex numbers discovered by Hamilton in 1843. Hyperkähler manifolds are remarkable for their ubiquity in high energy physics, quantum field theory, and string theory. They arise naturally as parameter spaces for Yang-Mills instantons, magnetic monopoles, Higgs bundles, and solutions of many other physical equations. The P.I. will study the geometry and topology of hyperkähler manifolds. He will construct new examples of hyperkähler manifolds, and/or their singular counterparts, hyperkähler orbifolds. He will also establish new restrictions on the possible topological types of hyperkähler manifolds. This project will contribute to a better understanding of a class of geometric spaces that lie at the heart of many physical models, and which also connect different areas of mathematics including algebraic and differential geometry, topology, and number theory. The P.I. will mentor PhD, masters, and undergraduate honors students, who will assist with the research project. He will enhance the training opportunities available to graduate students at the University of North Carolina by organizing mini-schools on advanced topics, by promoting student-led seminars, and by modernizing the geometry and topology courses. At the undergraduate level he will lead a problem solving seminar to coach students for mathematics competitions, facilitate research through honors projects, and initiate a new study abroad summer program for math majors and potential math majors. He will advocate for diversity by actively recruiting first generation college students and students from other under-represented groups to participate in these non-traditional activities.
The structure of hyperkähler manifolds, and their applications in physics, are well studied, yet only few compact examples are known: just two or three deformation classes in each dimension. At the same time, it is not known how many deformation classes there might be in each dimension. The P.I. is motivated by the problem of showing that this number is finite. He aims to show that every hyperkähler manifold can be deformed to a Lagrangian fibration, a hyperkähler manifold admitting a holomorphic fibre space structure. He then plans to establish general finiteness results by refining his earlier results for Lagrangian fibrations. He will exploit the analogies between compact and non-compact Lagrangian fibrations, such as Hitchin systems, to find new examples. The P.I. will also demonstrate general topological bounds on hyperkähler manifolds by exploring the structure of the cohomology ring. The ultimate goal is a more complete understanding of the possible topologies of hyperkähler manifolds.
