Poisson and Dirac geometry
Poisson structures and Dirac geometry, including deformation theory, shifted symplectic geometry, and reduction.
Overview
Research interests
My research concerns Poisson geometry, Dirac structures, symplectic and hyperkähler geometry, deformation theory, and moduli spaces. Two of my papers address these themes directly: From multiplicative to additive geometry: Deformation theory and 2D TQFT develops the Poisson-geometric passage between Hamiltonian and quasi-Hamiltonian structures, while Dirac geometry, deformation theory and shifted symplectic geometry develops the Dirac-geometric side.
A recurring question is how additive and multiplicative structures relate: Hamiltonian spaces carry Lie-algebra-valued moment maps, while quasi-Hamiltonian spaces carry group-valued moment maps. I study deformations between these settings and their behavior under fusion, reduction, and gluing.
Another part of my work builds geometric field theories from moduli spaces associated with quivers. My recent research develops a hyperkähler analogue using Nahm--Kirchhoff equations and analytic variational methods.
Current directions
Research themes
Quiver moduli and TQFT
Lax--Kirchhoff equations, Hamiltonian and quasi-Hamiltonian moduli spaces, and cutting-and-gluing structures.
Hyperkähler geometry
Nahm--Kirchhoff moduli spaces, hyperkähler quotients, and analysis in gauge-theoretic moduli problems.
Publications and preprints
Selected papers
Nahm--Kirchhoff moduli spaces, hyperkähler quotient and 2D field theories
Abstract
This work develops hyperkähler moduli spaces associated with Nahm--Kirchhoff equations on quivers and studies their two-dimensional field-theoretic structure. Analytic and variational methods prove a quiver analogue of a Donaldson-type theorem.
Dirac geometry, deformation theory and shifted symplectic geometry
Abstract
This work extends deformation methods to Dirac geometry and studies compatibility with reduction along strong Dirac maps, integration to quasi-symplectic groupoids, and Morita equivalence of Lie algebroids.
From multiplicative to additive geometry: Deformation theory and 2D TQFT
Abstract
This paper studies the Poisson-geometric transition between the multiplicative, group-valued setting of quasi-Hamiltonian spaces and the additive, Lie-algebra-valued setting of Hamiltonian spaces. It constructs deformations between these geometries and relates them to moduli spaces and two-dimensional field theories.
Lax--Kirchhoff moduli spaces and Hamiltonian 2D TQFT
Abstract
The paper constructs symplectic moduli spaces associated with Lax equations on quivers and Kirchhoff conditions at vertices. These spaces satisfy cutting and gluing laws and define a two-dimensional Hamiltonian TQFT.
Deformation spaces of quasi-Hamiltonian spaces
Abstract
This joint work constructs smooth deformations from quasi-Hamiltonian to Hamiltonian spaces, including deformations of the double, conjugacy classes near the identity, and moduli spaces of flat connections. Under suitable regularity assumptions, the construction is compatible with fusion and reduction.
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