Topics in Arithmetic Geometry and K-theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

This proposal concerns questions in higher dimensional Arakelov theory and algebraic K-theory. The PI intends to extend results of previous work with Soule on motivic weight complexes of algebraic varities to the arithmetic case. In collaboration with J. Hu, he intends to use the deformation to the normal cone techniquess for Arakelov theory (developed by J. Hu in his thesis) to the case of stacks. He also intends to study questions in arithmetic geometry related to the Mordell conjecture using methods from differential algebra and Arakelov theory. This proposal deals with arithmetic geometry and algebraic K-theory. Arithmetic geometry is the study of the properties of equations with coefficients that are whole numbers, and using methods both from number theory (the study of properties of whole numbers) and algebraic geometry, which studies geometric figures that can be defined by the simplest of equations, namely polynomials. Algebraic K-theory studies properties of linear equations using methods from geometry. The questions and phenomena which arise from combining number theory and geometry serve as driving forces in much of contemporary mathematics research. Moreover, arithemetic geometry has contributed to many applications including cryptography and theoretical computer science.