Introduction to semi-classical analysis

• Grant number: 25/23049-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: January 01, 2026
• End date: December 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Igor Mencattini
• Grantee: Silvio Lacerda de Almeida
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The research project aims to introduce the student to the study of semi-classical analysis, divided into four fundamental topics: Symplectic Geometry, Asymptotic Analysis, Symbolic Calculus, and Quantization. During the symplectic geometry section, the basic concepts of Hamiltonian mechanics will be studied, including the notion of the symplectic Poisson bracket structure and Hamiltonian vector fields. The asymptotic analysis section will be dedicated to the study of mathematical analysis concepts, that are necessary to study chapters focused on symbolic calculus, thus establishing the notions of distribution and the Fourier Transform of distributions - already encountered in Mathematical Physics course. An particular emphasis will be placed on the stationary phase theorem, due to the expansions and analyses it allows. The chapters related to symbolic calculus will be dedicated to introducing the symbol classes for pseudo-differential operators relevant to the last part of the project. This last part will be entirely devoted to the problem of quantization, so-called semi-classical. (AU)