Studies in K-theory and Arithmetic

University Of Utah, Salt Lake City UT

Investigators

Abstract

The investigator studies various K-theories of algebraic varieties and their relation to arithmetic cohomologies. In the case of classical algebraic K-theory this relation has found numerous applications in number theory, for example, to special values of L-functions of motives. In the last couple of year the investigator has introduced log-K-theory groups of log-schemes (log-scheme is a generalization of a toric variety) and studied their basic properties. She proposes to continue studying these groups as well as regulator maps into log-crystalline and log-etale cohomologies. She believes that this will have important arithmetic applications. This proposal deals with number theory and algebraic geometry. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. Algebraic geometry studies geometric figures that can be defined by the simplest of equations, namely polynomials. The questions and phenomena which arise from combining these two subjects serve as driving forces in much of contemporary mathematics research. Moreover, the combination of these subjects has contributed many applications in such diverse areas as codes and data transmission, robotics, and theoretical computer science.