The PI defines and studies invariants of geometric spaces and quantum analogues thereof. This area arises from the study of symmetries of geometric or physical systems and their linear actions (representation theory) and their quantization, the mathematical version of passing from classical to quantum mechanics (noncommutative geometry). There is a rich interplay between the two, which has connections and applications to many areas of mathematics, such as combinatorics, integrable systems, real algebraic geometry, quiver varieties, and resolutions of symplectic singularities.
The PI defines new homology theories which generalize de Rham cohomology and gives new interpretations of Hochschild and cyclic homology using D-module techniques. These ideas have applications to the representation theory of Lie groups, to the study of various algebras (Cherednik, symplectic reflection, and W-algebras), and to symplectic and Calabi-Yau resolutions. The PI will prove that his Poisson-de Rham homology recovers the de Rham homology of every symplectic resolution in new cases, such as for determinantal varieties and hypertoric varieties. He will recover from it important polynomials such as Kostka and Tutte polynomials. The PI plans to pursue conjectures relating this to the orders of vanishing of holomorphic fiberwise-closed forms on the deformation of the resolution. The main technique uses D-modules which encapsulate the Hamiltonian flow, built of canonical local systems on symplectic leaves. He also plans to use cyclic homology to obtain representations of affine Hecke algebras via the Gauss-Manin connection on noncommutative deformations of the mirror of cotangent bundles to flag varieties.
