The construction by Morel-Voevodsky of A^1-homotopy theory has enabled the systematic study of homotopy theory in the setting of algebraic geometry, as well as the application of techniques and approaches of homotopy theory to basic problems in algebraic geometry. We plan to pursue a research program in both aspects of this new theory: to continue our investigation of the basic structural issues of A1-homotopy theory and to use the new viewpoint introduced by this theory to shed light on basic problems in algebraic geometry, especially in the theory of algebraic cycles. Specific projects include:
1. Understanding Voevodsky's slice tower for naturally occuring cohomology theories, and constructing interesting cycle theories from these cohomology theories. These include non-oriented cycles arising from KO-theory, and an algebraic Bredon theory, arising from the K-theory of equivariant bundles.
2. Constructions of limit motives and limit motivic cohomology, with applications to actions of the Galois group of Q on geometric fundamental groups, and the study of periods of mixed Tate motives.
3. Using the algebraic cobordism of the moduli space of n-pointed genus zero curves to give interesting deformations of tensor categories.
4. Giving an algebraic setting for the study of stable phenomena in the moduli of smooth curves.
Creating analogies between the seemingly unrelated fields of algebra and topology has often been a fruitful approach to solving difficult problems in both fields. Recently, Morel and Voevodsky have transferred an entire branch of topology, called stable homotopy theory, to the algebraic setting, making ideas from stable homotopy theory applicable to problems in algebra and number theory. My research is an attempt to take a number of specific constructions from stable homotopy theory and adapt them to this new setting. In addition, I plan to make specific computations to determine what these new constructions yield when applied to familiar algebraic objects. Finally, I hope to answer basic questions in Galois theory and to explain why certain special numbers, the so-called multiple zeta values, always turn up when one evaluates the integrals known as periods of mixed Tate motives over the integers.
