Global Questions in Lie Groupoid Theory

该奖项是作为2021财年数学和物理科学上升博士后研究奖学金，MPS-Ascend计划的一部分。Joel Villatoro被授予该奖学金，在赞助科学家唐翔教授的指导下，在圣路易斯的华盛顿大学进行数学科学的研究和教育计划，包括其他学科的应用。 维拉托罗将调查几个相关的突出问题的理论李群胚和李代数胚。这些问题涉及几何对象的无穷小（小尺度）对称性与全局（大尺度）对称性之间的关系。李群胚是一种数学形式，可以用来模拟具有无限维的几何对象（即，非常大的）对称集。 沿着这项研究，维拉托罗将致力于为圣路易斯的贫困拉丁裔K-12学生制作教育数学材料。该项目的研究目标是研究李群胚理论中的问题。其中一些与李群胚作为奇异空间模型的研究一样古老。全纯辛群胚因其与代数几何和弦理论的关系而引起人们的兴趣。黎曼广群与线性化问题和紧性研究有关。主要研究者将需要使用来自不同领域的工具，包括微分几何，范畴理论，代数几何，群论，以及分析。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This award is made as part of the FY 2021 Mathematical and Physical Sciences Ascending Postdoctoral Research Fellowships, MPS-Ascend Program. Joel Villatoro is awarded this fellowship to conduct a program of research and education at Washington University in St. Louis in the mathematical sciences, including applications to other disciplines, under the mentorship of the sponsoring scientist Prof. Xiang Tang. Villatoro will investigate several related outstanding questions in the theory of Lie groupoids and Lie algebroids. These questions concern the relationship between infinitesimal (small scale) symmetries of geometric objects and global (large scale) symmetries. Lie groupoids are a mathematical formalism that can be used to model geometric objects with an infinite dimensional, (i.e., very large) set of symmetries. Along with this research, Villatoro will work on producing educational mathematics material for underprivileged Latino K-12 students in St. Louis.Research objectives of the project are to study problems in Lie groupoid theory. Some of them are as old as the study of Lie groupoids as models for singular spaces. Holomorphic symplectic groupoids are of interest due to their relationship to algebraic geometry and string theory. Riemannian groupoids are related to linearization problems and the study of compactness. The principal investigator will need to employ tools from various fields including differential geometry, category theory, algebraic geometry, group theory, as well as analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.