Algebraic Cycles On Splitting Varieties

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The project covers a wide range of aspects in algebra such as algebraic geometry, algebraic groups and motivic cohomology. The investigator proposes to use theory of algebraic cycles in geometry to study properties of splitting varieties of algebraic objects. The first topic the investigator proposes to study is the canonical dimension of an algebraic object such as algebraic variety, quadratic form, central simple algebra, etc. The canonical dimension measures complexity of the class of splitting fields of an algebraic object. In particular, the investigator proposes to compute the canonical dimension of projective homogeneous varieties and simple algebraic groups. The investigator also plans jointly with A. Suslin to compute the motivic cohomology of Severi-Brauer varieties and generalize this computation to the case of generic splitting varieties of arbitrary symbols. These varieties were used in the proof of the Milnor and the Bloch-Kato Conjecture. The main objective of mathematics is to provide an approximation to the picture of the physical world. This project develops methods from topology that studies continuous transformations of structures called topological spaces and algebraic geometry concerning geometric objects coming from graphing polynomial equations and called algebraic varieties. This project is devoted to the study of certain fundamental problems of motivic cohomology theory - a relatively new and very quickly developing branch of algebraic geometry. The new areas of algebra that will hopefully evolve from the work on the project will create new research opportunities for mentoring graduate students and junior faculty and provide material for graduate courses.