The PI will pursue several projects concerning the study of algebraic cycles, motives, and K-theory. In particular, he intends to employ the morphic Abel-Jacobi map, defined using Lawson homology, to better understand the group of cycles on a smooth, complex variety that are algebraically equivalent to zero and to better understand the behavior of Griffiths' Abel-Jacobi map on such cycles. (Some of this research may be carried out in collaboration with Andreas Rosenschon.) For example, the PI seeks to use the morphic Abel-Jacobi map to find counter-examples to the so-called "universality" of Griffiths' Abel-Jacobi map and to better understand the torsion subgroup of the Chow group. A related project will employ Lawson homology, morphic cohomology, and the morphic Abel-Jacobi map to better understand the (still hypothetical) category of mixed motives. The PI will also study cycles on and the K-theory of real algebraic varieties. One goal in this direction is to define and study higher algebraic K-groups for real varieties. These groups ought to be real algebro-geometric invariants, depending only on the set of real points cut out by a system of equations. Another project concerns developing a morphic cohomology analogue of 1-motives and 2-motives. Finally, the PI, in collaboration with Christian Haesemeyer and Eric Friedlander, hopes to extend the definitions of morphic cohomology and Lawson homology to ground fields other than the real and complex numbers.
An algebraic variety is a geometric object defined by a collection of polynomial equations. For example, one can define a circle using one such equation involving two variables, but, in general, an algebraic variety is given by any number of equations in possibly a great many variables, and the resulting geometric object might not fit inside our usual three-dimensional world. A basic question in algebraic geometry is whether two given systems of polynomials equations actually determine the "same" (i.e., isomorphic) algebraic variety. Ideally, one would like a list of invariants that uniquely determine when two varieties are isomorphic, but despite its apparent reasonableness, this goal is completely unrealistic. Rather, one hopes to find invariants that classify broad categories of varieties in some geometrically meaningful way. Algebraic cycles, K-theory, and motivic cohomology are examples of such invariants, and there have been tremendous advances in recent years in our understanding of these fundamental invariants. Perhaps the most famous one is Voevodsky's proof of the Beilinson-Lichtenbaum conjecture at the prime 2, for which he was awarded the Fields Medal in 2002. For all the recent successes, there is still much we do not understand about these basic invariants, even for smooth, projective complex varieties. The Hodge conjecture (which is one of the seven Millennium Problems sponsored by the Clay Mathematics Institute) and Bloch's conjecture concerning zero-cycles on surfaces, both of which remain unsolved, are two such examples. The PI will pursue several research projects aimed at further understanding the fundamental invariants of algebraic varieties, typically focusing on smooth, projective complex varieties. Most of his projects focus on the study of cycles on such varieties, often by using the relatively new invariants of Lawson homology, morphic cohomology, and the morphic Abel-Jacobi map.
