Hodge Type Realizations of Algebraic Cycles

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An abridged title description of my research is Regulators of Motives. The world of motives was the invented by A. Grothendieck, a Fields medalist who is arguably the most influential mathematician in the 20th century. Grothendieck observed that many different mathematical objects, arising from very different origins (e.g. algebra, analysis, or geometry) had more in common than what was formerly understood. He conjectured that there was an entirely complete mathematical world in itself, called the category of motives, that would unify and encode all of the deep ideas in mathematics, and in a somewhat universal way. This world of motives, which is still currently in conjectural form has been a source of preoccupation among the leading mathematical minds of the 20th and 21st centuries. A case at point is a candidate version of motives possessing ``most'' of the desired properties, invented by 2004 Fields medalist V. Voevodsky. Regulators are realizations of the conjectural world of motives into categories that are more ``earthly'' defined. So in a sense, regulators gives us a snapshot of something we are trying to show exists! Of course when we construct a regulator, we have a candidate category of motives in mind. The trouble is that outside of Grothendieck's original concrete proposal (which requires major conjectures to get off the ground), no other candidate possessing the expected properties is very easy to describe by itself, and that is precisely where regulators come into play. Although there are plenty of very abstract definitions of regulators in the literature, the absence of a very explicit description was a big obstacle. Over the course of the last 10 years, I took it upon myself with various collaborators to provide that description in a first paper (with Kerr and M\"uller-Stach, Compositio Math 142). A second paper (with Kerr, Inventiones 170), which was much more involved, was a tour de force, ``take no prisoners'' approach to regulators involving the very general arena of ``mixed motives'', where applications to number theory and physics are apparent. The next paper in this direction which is currently in progress, and using a larger cast of collaborators, will be a explicit and decisive description of regulators using Bloch's simplicial higher Chow groups as well as using the Voevodsky machinery. Finally, Kerr and Lewis have reworked a simplicial version of the Bloch regulator, and in the process, have discovered a serious error in the simplicial real regulator provided by Goncharov, which has been extensively used over the past decade. The results will appear in the Journal of Algebraic Geometry. The corollaries to all this work will hopefully influence a new generation of algebraic geometers for many years to come. My current preoccupation is in the direction of the Beilinson-Hodge conjecture, and its connections to the Bloch-Kato theorem, as well as work on height pairings.