Global Questions in Lie Groupoid Theory

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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.