Algebraic K-theory and homology of linear groups

• Grant number: 15/20361-3
• Support Opportunities: Regular Research Grants
• Start date: February 01, 2016
• End date: January 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Behrooz Mirzaii
• Grantee: Behrooz Mirzaii
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

In this project we will study Algebraic K-theory of rings and its relations, interactions and applications to problems in Algebra, Algebraic Topology and Hyperbolic Geometry. Our plan is divided into two very closely related parts:(1) Algebraic K-theory of rings, (2) Homology of Linear GroupsThe Hurewicz map in algebraic topology relates K-groups of a ring to the homology of the stable general linear groups of the ring, which is an important tool to investigate K-groups. The study of parts (1) and (2) through the Hurewicz map is at the center of our project.This method of studying K-groups already has shown its strength, for example Quillen's proof that K-groups of the ring of integers of number fields are finitely generated, his calculation of the K-groups of finite fields, Suslin's proof of Quillen-Lichtenbaum conjecture on the torsion of the K-groups of algebraically closed fields, etc. 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 this field. (AU)