Research Projects

The project is in the general area of motivic homotopy theory and motivic cohomology, and their application to questions in algebraic geometry and algebraic groups. The general idea of motivic homotopy theory is to take concepts and methods from classical homotopy theory and adapt them in a suitable manner in the algebraic context, whenever possible. The foundational work of Morel, building on the work of Morel and Voevodsky, emphasizes that the relationship between motivic homotopy theory over a field and the category of smooth schemes over that field is analogous to the relationship between classical homotopy theory and the category of smooth manifolds. In this project, we plan to study several motivic invariants such as motivic homotopy sheaves, cellular motivic homology and motivic cohomology and their applications to questions in algebraic geometry. We aim to provide new connections and tools for the study of long standing and important questions in algebraic geometry involving near-rationality properties of algebraic varieties, R-equivalence in algebraic groups, Rost nilpotence for algebraic cycles and etale cohomology with finite coefficients.