Homology of linear groups over Dedekind domains

• Grant number: 18/03561-7
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: December 01, 2018
• End date: November 30, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Daniel Levcovitz
• Grantee: Fatemeh Yeganeh Mokari
• Supervisor: Kevin Hutchinson
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: University College Dublin, Ireland
• Associated to the scholarship: 16/13937-9 - Valuation theory of group rings and homology of soluble groups, BP.PD

Abstract

Linear groups are the main source of many fundamental ideas in mathematics. They appear in almost all subjects of mathematics and have many algebraic and geometric properties. Linear groups are groups that are isomorphic with matrix groups, i.e. a group of invertible matrices over a commutative ring with unit. Matrices are very powerful tool for computation. The principal objective of this project is to solve some fundamental problems of the homology of linear groups over Dedekind domains using tools from the theory of groups, theory of numbers and algebraic K-theory. Dedekind domains are natural generalization of the ring of integers. The ring of algebraic integers of a number field is a classical example of a Dedekind domain. Homology of linear groups appear in many different areas of algebra and geometry. The study of these groups have been the subject of investigation for many years. Explicit computations of these groups have been hard and only a complete calculation of few cases have been achieved. This project is connected to three major branches of mathematics: Group Theory, Algebraic Number Theory and Algebraic K-theory. This project and its objectives are based on the most important questions and conjectures which have not been answered and have been the source of inspiration for some of the most outstanding works in the fields of Algebra, Number Theory and Geometry. (AU)