Universal Series, Chow Rings, and Dualities in the Moduli Theory of Sheaves

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This project is in the field of algebraic geometry which can be described as the study of geometric spaces cut out by solution sets of systems of polynomial equations. The subject has a strong classical tradition. At present, in one significant strand, research efforts in the field go toward establishing mathematical foundations for the physics of the early universe. A particularly important role in this effort is played by moduli theory, which deals with the classification and deformation properties of important geometric objects of the same type. The current project will investigate moduli theory of sheaves, objects corresponding to gauge fields (such as the electromagnetic field) in physics. Mathematical quantities associated with moduli spaces of sheaves correspond to amplitudes of scattering processes in particle physics. Studying the structure of these quantities therefore bears on important questions in both geometry and high-energy physics. Thus, while contributing to geometry, the project also seeks to improve our understanding of theoretical physics.In one direction, the PI will investigate central series of invariants on Hilbert schemes of points on surfaces -- spaces which have emerged as essential objects in both geometry and representation theory. She will also try to put on rigorous geometric footing a q-deformation of two dimensional Yang-Mills theory, by interpreting it in a four-dimensional context, and connecting it with recent work on modularity of invariants for moduli spaces of stable sheaves on surfaces. In a different direction, the PI's efforts will be concentrated on understanding the groups of cycles modulo rational equivalence of moduli spaces of stable sheaves on curves and K3 surfaces, not least with a view toward confirming the Hodge conjecture for these spaces.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.