Sub-Laplacians on compact Lie groups

• Grant number: 25/08151-5
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: July 01, 2025
• End date: June 30, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Agreement: Research Foundation - Flanders (FWO)
• Principal Investigator: Paulo Leandro Dattori da Silva
• Grantee: André Pedroso Kowacs
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 24/08416-6 - Global hypoellipticity and solvability on product manifolds, AP.R

Abstract

The hypoellipticity ($C^\infty$, analytic and Gevrey, both local and global) of sub-Laplacians has been addressed by several authors, but it is still a problem with many unresolved questions, since no necessary and sufficient conditions are known to characterize the hypoellipticity of a sub-Laplacian in its general form. We propose a new line of investigation: the global hypoellipticity and solvability of sub-Laplacians associated to involutive systems of real vector fields defined on compact Lie groups. The choice of a compact Lie group, either as the ambient manifold or, more generally, as a symmetry group for our PDEs, serves several purposes. On the one hand, it is more general than the torus; therefore, we hope to extend the results currently restricted to the torus to a more general setting of a compact Lie group $G$. On the other hand, it is less general than an abstract compact manifold; our expectation here is to obtain more precise results and a more detailed description of phenomena already known on general manifolds where symmetries are absent. Thus, Lie groups promise to be a rich middle ground between these two extremes, potentially with additional applications in special cases, such as special unitary groups. (AU)