Heat kernel and analysis on riemannian manifolds

• Grant number: 21/09540-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: November 01, 2021
• End date: October 31, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Alexandre Paiva Barreto
• Grantee: Gabriel Cardoso Pigozzi
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 16/24707-4 - Algebraic, geometric and differential topology, AP.TEM

Abstract

This project aims to develop an introductory study of the domain of Geometry called Geometric Analysis, whose value nowadays is unquestionable. The usage of analytical techniques in geometry not only provides a generalization of classical analysis, but also allowed important advances in the last century, such as the study of the Heat Kernel by S. T. Yau and contributions on geometric fluxes by R. Hamilton, that were fundamental in the proving of the Poincaré Conjecture by G. Perelman. In this project, we will introduce the student to several important results in Geometric Analysis, emphasizing the study of function spaces, differential operators and their weak extensions, in particular techniques about the Heat Kernel, both in Euclidean spaces and in abstract manifolds. (AU)