Poisson Geometry, Quantum Moduli, and Geometric Dualities

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This is a research in algebraic geometry. The field of algebraic geometry studies geometric models by distilling and encoding their essential complexity in polynomial equations. The project integrates ideas from quantum physics and the physical study of symmetries of the fundamental laws of nature to extract new and unexpected information about the geometry of spaces. The project will focus on unravelling the hidden structure of multi-dimensional geometric models of quantum fields and to capture this structure in a cascade of polynomial invariants. These invariants will give a new mathematical tool for understanding and proving various empirically observed physics dualities which are expected to identify a priori unrelated quantum theories. The project aims to unify the analytic and geometric properties of parameter spaces of representations in arbitrary dimension and sets the stage for understanding the basic structure of moduli problems in a way suitable for pragmatic use in a broad spectrum of applications. The proposed work will be immediately relevant to deep questions in symplectic geometry, geometric representation theory, string theory and quantum field theory.Three directions will be studied. The first is to investigate how the Hodge theory of varieties with potentials and the associated perverse and weight filtrations are exchanged by T-duality. In the second project a new method will be developed for constructing moduli of irregular connections on non-compact algebraic manifolds, for building explicit symplectic foliations, and for computing their leaves. This will involve a new de Rham theory of formal boundaries of varieties, and a new construction of sheaf theoretic invariants from the formal geometry. The final project will build a spectral coverformalism for constructing Hecke eigensheaves on moduli of bundles related to intersections of quadrics.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.