Reading Course on the Theory of Measure and Integration and Introduction to the Geometric Measure Theory.

• Grant number: 23/12997-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: November 01, 2023
• End date: October 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Hermano Frid Neto
• Grantee: Henrique Casellato Vitorio Rodrigues da Costa
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil

Abstract

Measure Theory is a basic tool of mathematical analysis. It has become extremely important in several branches of mathematics since its foundation in Henri Lebesgue's doctoral thesis in 1902, following previous contributions made by René-Louis Baire, Camile Jordan, and Émile Borel, among others, the latter being his advisor. A famous example of its impact was its application in the establishment of the axioms of Probability Theory by Andrey Kolmogorov in 1933. Other fundamental developments occurred, among many others, in the study of partial differential equations, fractal geometry, ergodic theory, and the study of chaotic systems. In this project, we intend that the student will be able to, after acquiring a basic knowledge of the main concepts in measure theory and integration, begin studying one of its most important applications, namely, the geometric theory of measurement, learning basic notions such as the Hausdorff measure, Hausdorff dimension, area, and co-area theorems and their application to the classic Plateau problem of minimal surfaces.