B.A. Haverford College

Ph.D. University of Chicago

Associate Professor

Areas of Expertise: Algebraic Geometry, especially Algebraic Cycles

Research Summary:

The basic problem in Algebraic Geometry is to describe the solution sets of systems of polynomial equations. As a simple example consider the equation y2 + y - x3 + x = 0. The solution set (with point at infinity added) in real numbers x and y consists of two circles. If x and y are allowed to be complex numbers, the solutions form the surface of a donut. If x and y must be rational numbers, we obtain an infinite set with a natural group structure isomorphic to the integers. Finally, if x and y must be integers, we are asking when the product of two consecutive integers equals the product of three consecutive integers. This occurs only finitely many times. The notion of scheme encodes all this information into a single mathematical object and allows one to use knowledge about the solution set for one number system to deduce information about solution sets for other number systems.

One way to study schemes is by associating to them a number of algebraic ``models''. The idea is that the relatively simple models should detect fundamental properties of the complicated schemes. In fact some of these models such as the cohomology ring are now quite well understood and provide practical computational tools. Others, such as the Chow group, which is constructed using all the subvarieties of the given scheme and is aimed at detecting subtle properties that singular cohomology can not see, remain mysterious. I study the Chow group by trying to understand how it relates to the better understood algebraic invariants. The relationship with l-adic Galois modules and Hodge structures, is a focal point of my research (1,2,3,7,8,9,10). Questions about Chow groups lead to a broad range of geometric problems (4,5,6,11).

Selected Publications:

1. Algebraic cycles on certain desingularized nodal hypersurfaces, Math. Ann. 270, 17-27 (1985).
2. On complex multiplication cycles on elliptic modular threefolds, Duke Math. Journal 53, 771-794 (1986).
3. Hodge classes on self-products of a variety with an automorphism, Compositio Math. 65, 3-32 (1988).
4. On fiber products of rational elliptic surfaces with section, Math. Z. 197, 177-199 (1988).
5. On certain modular representations in the cohomology of algebraic curves, J. of Algebra 135, 1-18 (1990).
6. Bounds for rational points on twists of constant hyperelliptic curves, J. fuer d. reine u. angew. Mathe. 411, 196-204 (1990).
7. Some examples of torsion in the Griffiths group, Math. Ann. 293, 651-679 (1992).
8. Complex multiplication cycles and a conjecture of Bloch and Beilinson, Trans. of A.M.S. 339, 87-115 (1993).
9. On Hodge structures and non-representablity of Chow groups, Compositio Math. 88, 285-316 (1993).
10. On the computation of the cycle class map for nullhomologous cycles over the algebraic closure of a finite field, Anal. Scien. École Norm. Sup. 4e série, t. 28, 1-50 (1995).
11. Varieties dominated by product varieties, Int. J. of Math. 7, 541-571 (1996)