Groups, representations, and applications

• Grant number: 19/17085-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): October 01, 2019
• Effective date (End): September 30, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Gabriel Guzatti Vilas Boas
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/23690-6 - Structures, representations, and applications of algebraic systems, AP.TEM

Abstract

In this research project, we aim at studying some of the basic notions of Contemporary Algebra. We are commencing with a brief revision of Group Theory (the student has already studied the basics of the area, and he did a formal study as a Introduction to Science, and moreover, now he is studying topics of Group Theory). Afterwards we will continue with the Representation Theory of finite groups. Our approach will be through the structure of the group algebra of the corresponding group this will require notions from the theory of rings and algebras (the radical, semisimplicity). Next, we shall study the theory of representations of the symmetric group, the so-called Young theory. Such studies require quite a deep knowledge of Algebraic Combinatorics. All this will take approximately 1 year. Depending on Gabriels interests we could continue, after the first year, either with applications of Representation Theory to the theory of algebras satisfying polynomial identities, or with the Invariant theory of classical groups. Or else some other topic in Algebra (depending on Gabriels interests). The student has already studied some of the notions: he did an IC on Group Theory before coming to the State University of Campinas. Now Gabriel receives a scholarship through the PICME program as a winner at the Brazilian Math Olympiads for the secondary school. Now he is studying, under my supervision, some additional topics in Group Theory and Applications. Without any doubt, Gabriel has the conditions to continue, after graduating, with post-graduate studies in Mathematics. Thus the study of the topics covered in this project will contribute significantly to form a better culture and deeper knowledge in several areas of Maths, more specifically Algebra and Algebraic Combinatorics. Gabriel is among the best students in his class (as it can be seen from his academic records). In addition to the courses taken in Campinas he already did various courses at University level. He is mature enough to make good use of the scholarship.