Mathmatical analysis, differentiation, Lebesgue integral and functional analysis

• Grant number: 17/26382-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2018
• End date: March 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Hildebrando Munhoz Rodrigues
• Grantee: João Vitor Nazari Formagio
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The main objective of this project is the study of Mathematical Analysis, with the purpose of deepening the knowledge in Mathematics of the students and giving them a solid basis for better development of their academic and professional lives. The seminars will be presented in English. In addition to introducing numerical sets and their properties, we will study basic notions of Metric Spaces Topology, Continuous Functions, Differentiable Functions, Numeric Sequences and Series of Numeric Functions. Then we will study measurement and integration theory, more specifically the Lebesgue integral and its applications, such as integrals of functions dependent on parameters and derivation under the integration signal. The books studied will be Walter Rudin's Principles of Mathematical Analysis and C. S. Hönig's "The Integral of Lebesgue and its Applications. "This entire program will be developed in one year. This will serve as the basis for a second program to be started where topics of operator theory in finite-dimensional vector spaces ranging from the basics of vector spaces and normed vector spaces to positive matrices will be studied. In this module will be used mainly the book "A Short Introduction to Perturbation Theory for Linear Operators" by T. Kato. We will use other reference books if it is convenient, whose relationship is presented in bibliographical references. As a complement, participants should familiarize themselves with AMS-LATEX software, which will be used for reporting.