Global Questions in Lie Groupoid Theory

该奖项是2021财年数学和物理科学提升博士后研究奖学金(MPS-Ascend Program)的一部分。Joel Villatoro被授予这一奖学金，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开展数学科学的研究和教育项目，包括在其他学科的应用。维拉托罗将研究李群胚和李代数体理论中几个相关的悬而未决的问题。这些问题涉及几何对象的无穷小(小尺度)对称性和全局(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(即，非常大的)对称集的几何对象。除了这项研究，维拉托罗将致力于为圣路易斯市贫困的拉丁裔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。他们中的一些人和研究李群胚一样古老，也是奇异空间的模型。全纯辛群群因其与代数几何和弦理论的关系而引起人们的兴趣。黎曼群胚与线性化问题和紧性研究有关。首席研究员将需要使用不同领域的工具，包括微分几何、范畴理论、代数几何、群论以及分析。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.