Motivic Cohomology, Motivic Homotopy Theory and K-theory

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This award supports the principal investigator's research on the connections between Algebra, Algebraic Geometry, and Topology. The goal is to gain a better understanding of the way that structures in algebraic geometry are reflected by structures in topology, using a blend of algebraic and topological methods. Several old conjectures will be attacked, and hopefully settled. In addition the project provides research training opportunities for graduate students.One part of the project is to find clean proofs of parts of the proof of the recently-verified Bloch-Kato conjecture. A related part of the project involves motivic homotopy theory; one goal is to understand how Voevodsky's slice filtration is related to algebraic cobordism and other motivic spectra and to verify several of the slice conjectures. The PI will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques. A final part of this project is to show that the algebraic Witt group of varieties behaves well with respect to polynomial and Laurent polynomial extensions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.