报告题目/Title
Desingularizing singular symplectic structures
报告人/Speaker
Eva Miranda (Polytechnic University of Catalonia)
报告时间/Time
2024年10月18日 20:00-22:00
报告地点/Venue
Zoom ID: 904 645 6677，Password: 2024
会议链接/Link
报告摘要/Abstract
The exploration of symplectic structures on manifolds with boundaries has naturally led to the identification of a “simple” class of Poisson manifolds. These manifolds are symplectic away from a critical hypersurface, but degenerate along this hypersurface. In the literature, they are referred to as b-symplectic or log-symplectic manifolds. They arise in the context of the space of geodesics of the Lorenz plane and serve as a natural phase space for problems in celestial mechanics such as the restricted 3-body problem. Geometrically, these manifolds can be described as open symplectic manifolds endowed with a cosymplectic structure on the open ends.
The technique of "deblogging" or desingularization associates a family of symplectic structures to singular symplectic structures with even exponent (known as b^{2k}-symplectic structures), and a family of folded symplectic structures for odd exponent (b^{2k+1}-symplectic structures). This method has good convergence properties and generalizes to its odd-dimensional counterpart, contact geometry. In this way, the desingularization technique puts under the same umbrella various geometries, such as symplectic, folded-symplectic, contact, and Poisson geometry.
The desingularization kit has a broad range of applications, such as the construction of action-angle coordinates for integrable systems, KAM theory, quantization, and counting periodic orbits.
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