Analysis, Algebra and Applications

Abstract

In this research project for Introduction to Research we shall study the fundamental algebraic structures: groups, rings, fields, and their properties. We shall also study basic notions of Analysis. We shall use the books by M. Bre\v sar, Undergraduate Algebra, and by W. Rudin, Principles of Analysis. These are excellent choices of books which treat in a rigorous and didactic way the fundamentals of Algebra and of Analysis. The student is enrolled in the specialty of Computer Science; the courses in Mathematics he is supposed to do comprise the three Calculis, Analytic Geometry, and Linear Algebra. The student has excellent academic record. He was among the best students at the Brazilian Mathematical Olympiads scoring several silver medals. Here we recall that the Brazilian Math Olympiad is, without any doubt, the most selective and rigorous Scientific Olympiad: it distributes only 5 to 6 gold and up to a dozen silver medals per year for secondary school students. In the last edition, 2022, of the International Competititon in Mathematics for University Students, organized by the University College, London, Yvens got silver medal. Thus the student has proven record of capacity, competence, and ability in studying Mathematics. We shall aim at forming a good and ample mathematical culture, with solid knowledge at least in those two fundamental subareas of Mathematics. During a possible extension for a second year of this Introduction to Research we can choose a specific topic to work on, depending on the student's preferences, and we can expand the student's skills and knowledge on such a topic. Therefore the main goal of this project for Introduction to Research will be the study of the basics of Mathematics.