NSF Award
1811995
In many areas of geometry and physics, the collection of objects under investigation can be organized together to form a space with intrinsic geometry, known as a "moduli space". Moduli spaces may parameterize a wide variety of objects, from physical particles, to geometric shapes, to solutions of differential equations. They are important not only because they carry information about the objects they parameterize, but also because they represent naturally occurring examples of interesting geometries. There are several well-known families of moduli spaces which carry a so-called hyperKahler geometry, based on the algebra of quaternions, and physical considerations suggest that these moduli spaces may have certain strong properties reflected in their global shape, or topology. A principal goal of this project is to develop analytical tools to establish these properties and more deeply understand the geometry of these spaces and their relatives. This work has significant interdisciplinary appeal as it makes contact with several different areas of analysis, geometry and topology, and provides a testing ground for certain duality principles in physics. Additional impacts of the project include the facilitation of the PI's educational activities, including the supervising of original undergraduate research, mentoring student-led project-based independent study projects, and an ongoing initiative to improve diversity and gender parity in the mathematics program at New College of Florida.<br/><br/>This project will develop a framework for the analysis of elliptic operators on a class of spaces which includes several families of hyperKahler moduli spaces of interest in physics and geometry. These are exemplified by the moduli spaces of magnetic monopoles, the L2 cohomology of which is the subject of a long standing and still open conjecture of Ashoke Sen coming from supersymmetric physics. This seminal conjecture spawned several parallel conjectures about similar moduli spaces, and is the starting point for a number interesting geometric questions motivated by physics. A principal tool this project will develop is a generalization of Mazzeo and Melrose's well-known calculus of pseudodifferential operators on manifolds with "fibered boundary", to a category of compact manifolds with "quasi-fibered boundary". This is a category in which the monopole moduli spaces and other families of hyperKahler moduli spaces admit natural compactifications, the construction of which is a second major goal of this project. Beginning with a proof of Sen's conjecture, this work will therefore make refined techniques of geometric microlocal analysis newly available to the study of these important classes of spaces in hyperKahler geometry and beyond.<br/><br/>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.
