The Symplectization Functor

纤维化近辛流形中某些哈密顿系统的辛化

Symplectization of certain Hamiltonian systems in fibered almost-symplectic manifolds

本文研究近辛流形上部分与完全哈密顿向量场的区别，证明非垂直完全哈密顿向量场在一般性条件下可约化为标准辛哈密顿向量场族。

Functoriality for symplectic and contact cutting, and equivariant radial-squared blowups

We exhibit Lerman's cutting procedure as a functor from the category of manifolds-with-boundary equipped with free circle actions near the boundary, with so-called equivariant transverse maps, to the category of manifolds and smooth maps. We then apply the cutting procedure to differential forms that are not necessarily symplectic, to distributions that are not necessarily contact, and to submanifolds. We obtain an inverse functor from so-called equivariant radial-squared blowup.

Integration of Lie Algebroid Comorphisms

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.

Double Groupoids and the Symplectic Category

We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on the fact that one can realize the core of a symplectic double …

Double Groupoids, Orbifolds, and the Symplectic Category

Motivated by an attempt to better understand the notion of a symplectic stack, we introduce the notion of a symplectic hopfoid, which should be thought of as the analog of a groupoid in the so-called symplectic category. After reviewing some foundational material on canonical relations and this category, we show that symplectic hopfoids provide a characterization of symplectic double groupoids in these terms. Then, we show how such structures may be used to produce examples of symplectic orbifolds, and conjecture that all symplectic orbifolds arise via a similar construction. The symplectic st…

On 2d TQFTs whose values are holomorphic symplectic varieties

For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the diagonal action of G. We describe various properties of eta_G obtained via string-theoretic analysis. Mathematicians are urged to construct eta_G rigorously.