Global properties of systems of vector fields on compact Lie groups

• Grant number: 24/21562-1
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: May 01, 2025
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Paulo Leandro Dattori da Silva
• Grantee: Fernanda Martins Simão
• Supervisor: Michael Ruzhansky
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Ghent University (UGent), Belgium
• Associated to the scholarship: 23/07703-9 - Tube type structures on compact Lie groups, BP.DR

Abstract

Systems of vector fields arise as a local basis of an involutive sub-bundle $\mathcal{V}$ of the complexified tangentbundle $\mathbb{C}T\mathcal{M}$. Examples of involutive structures $(\mathcal{M},\mathcal{V})$ include foliations, complex structures, and CR structures.In this project we will investigate global properties of involutive structures defined on a smooth manifold $M$. Many results are obtained considering $M$ being the $n$-dimensional torus $\mathbb{T}^n\simeq\mathbb{R}^n/2\pi\mathbb{Z}^n$ and, among other things, by using extensively Fourier analysis.We are proposing a new line of investigation: global hypoellipticity and solvability of partial differential equations associated with involutive system of (complex) vector fields defined on compact Lie groups. The choice of a compact Lie group $G$ either as the ambient manifold, or, more generally, as a group of symmetries, for our PDEs, plays a few roles. On the one hand, it is more general than the torus $\mathbb{T}^n$; hence, one hopes to extend results currently restricted to the latter to a more general $G$. On the other hand, it is less general than an abstract compact manifold $M$; our expectation here is to obtain on $G$ finer results and more detailed description of phenomena already know on general $M$ where symmetries are absent.