Reflections on the Hodge Conjecture from an Arithmetic Geometer

Speaker 

James D. Lewis is Professor at University of Alberta. His current research interests are in regulators of (higher) algebraic cycles into Hodge cohomology theories (such as Beilinson's absolute Hodge cohomology). With his colleague Matt Kerr, together initially with Stefan Mueller-Stach, they provided over the course of 10 years an explicit description of the Beilinson-Bloch regulators in terms of polylogarithmic currents. His earlier work led to a survey book on the Hodge conjecture, which was well received by the algebraic geometry community.

Abstract

This talk is based on recent developments from Claire Voisin (acting on the advice of Vincent Maillot and Christophe Soule), as well directions in higher K-theory, due to myself and Rob de Jeu. When it comes to the Hodge conjecture in higher K-theory, we explore the connections between the expected injectivity of the Abel-Jacobi map for varieties defined over number fields (Bloch/Beilinson), and the Hodge conjecture for the same varieties.