带李代数胚的丛值多重辛结构的几何

多辛几何中的同伦共动量映射

Homotopy Comomentum Maps in Multisymplectic Geometry

本文研究多辛几何中群作用容许同伦共动量映射的条件，给出多辛球面上紧群作用的完整分类，显式构造泊松代数到扭李代数胚的高阶嵌入，并构造了体积保持微分同胚群在三维欧氏空间上的同伦共动量映射，揭示其与流体动力学和纽结理论的联系。

扭Courant r-Lie代数与非扭Courant (r+1)-Lie代数胚之间的$L_\infty$-态射

$L_\infty$-morphisms between twisted Courant $r$-Lie algebras and untwisted Courant $(r{+}1)$-Lie algebroids

针对Marco Zambon提出的高阶闭形式扭化Courant代数胚是否存在典范$L_\infty$-态射的问题，本文给出肯定回答，提出通用构造框架并阐明其底层结构，验证框架可适配相关已有代数间的态射。

Momentum sections in Hamiltonian mechanics and sigma models

We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on a higher dimensional manifold.

A Frame Bundle Generalization of Multisymplectic Momentum Mappings

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_VY$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field mome…

The van Est Map on Geometric Stacks

We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies differentiable stacks, Lie algebroids and homotopy theory. In Part 2 of this thesis we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the cohomology of a simple foliation $\mathcal{H}\to\mathcal{G}$. In Part 1 we generalize the functions that we can take cohomology of…

Vector Bundle Valued Differential Forms on $\mathbb{N} Q$-manifolds

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known r…

泊松几何中的移位拉格朗日结构

Shifted lagrangian structures in Poisson geometry

本文通过建立移位辛几何与经典泊松几何之间的新联系，利用2-移位辛群中的拉格朗日态射，发展了狄拉克结构与拟泊松流形积分的理论。

Integration of twisted Dirac brackets

The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions. In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form. More generally, given a Lie groupoid $G$ over a manifold $M$, we show that multiplicative 2-forms on $G$ relatively closed with respect to a closed 3-form $ϕ$ on $M$ correspond to maps from the Lie algebroid of $G$ into the c…