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Key research themes
1. How do symplectic quotients and singular symplectic varieties arise from group actions and what are their singularity properties?
This theme investigates the construction of symplectic quotients by Lie group actions on symplectic manifolds or modules, focusing on their geometric and singularity properties. It addresses when such quotients admit symplectic singularities, graded Gorenstein structures, and rational singularities, which are crucial for understanding their geometric regularity and deformation theory. The works explore real and complex symplectic quotients, the role of module largeness conditions (such as k-largeness), and the relationships between symplectic quotients and orbifolds or singular symplectic varieties.
Key finding: Proves that for a 3-large unitary module (V,G) of a compact Lie group K with complexification G, the associated complex symplectic quotient μ⁻¹(0)//G has symplectic singularities and is graded Gorenstein; this extends to the... Read more
Key finding: Although not a direct study of symplectic quotients by Lie groups, this paper connects symplectic structures to algebraic number fields by characterizing number fields Q(λ) that admit symplectic forms invariant under... Read more
2. How do symplectic and Hamiltonian structures manifest in the geometry of polygon moduli spaces and Cartan–Hartogs domains?
This theme studies symplectic geometric structures arising in moduli spaces of polygons with fixed side lengths in Euclidean space and in special complex domains like Cartan-Hartogs domains. It focuses on the construction of symplectic quotients, Hamiltonian flows induced by geometric operations (such as bending flows in polygon spaces), integrable systems built from action-angle variables, and the existence of global symplectic coordinates and dualities in complex domains. These insights link the symplectic geometry of spaces with rich geometric and complex analytic structures, providing explicit constructions of symplectomorphisms and integrable Hamiltonian dynamics.
Key finding: Demonstrates that the moduli space M_r of polygons with fixed side lengths r in Euclidean space admits a natural symplectic structure realized as a weighted symplectic quotient of (S²)^n by the diagonal SU(2) action, and... Read more
Key finding: Constructs global Darboux coordinates (global symplectic coordinates) for Cartan–Hartogs domains and their dual domains, showing these domains admit symplectic duality if and only if they reduce to complex hyperbolic spaces.... Read more
3. How is symplectic geometry integrated with Poisson structures, singularity theory, and higher-dimensional birational operations?
Research in this theme explores the interaction between symplectic and Poisson geometry, especially in degenerate or singular contexts, including near-symplectic 4-manifolds and the symplectic behavior under birational modifications such as blow-ups and blow-downs in dimension 6. It addresses the existence and classification of singular Poisson structures arising from degenerations of symplectic forms, computes Poisson cohomology illuminating deformation and rigidity properties, and investigates criteria for symplectic blowing down involving exceptional divisors and fibration structures. These studies connect analytic, topological, and algebraic aspects of singular symplectic structures and birational geometry in dimensions relevant for modern enumerative and deformation theories.
Key finding: Constructs a singular Poisson structure π of maximal rank 4 on the tubular neighborhood of the singular locus Z_ω of a near-symplectic 4-manifold (M, ω), where π vanishes on a set containing Z_ω. Computes the smooth Poisson... Read more
Key finding: Provides criteria for symplectic blowing down in dimension 6, distinguishing cases of exceptional divisors diffeomorphic to symplectic P² with normal degree -1, which can always be blown down symplectically, from the more... Read more
Key finding: Develops algebraic methods to obtain formal normal forms of ω-Hamiltonian vector fields under semisymplectic Lie group actions incorporating symmetries and reversing symmetries, preserving the Hamiltonian structure and... Read more