Algebraic structures and applications

• Grant number: 22/05183-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2022
• End date: June 30, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Gabriel Ribeiro Paiva
• 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

The algebraic structures, namely groups, rings, fields, modules, and algebras, are of major importance and a must in any serious mathematical research. They also have direct, extremely important and immediate applications to Theoretical Physics, Chemistry, Computer Science, Linguistics, Logic, and many other branches of Science. In this research project we will start a detailed study of such structures. As a first goal the student will study in a profound way the fundamental notions of vector spaces, groups, rings, polynomials, fields, and algebras. Afterwards we will study Combinatorial Ring Theory. We will start with the basic notion of an action of a group on a set, and its properties. An immediate application will be the study of symmetric polynomials and their properties. This will allow us to make a smooth and natural transition to the notion of invariants of an action of a group. (AU)