This research project will contribute to algebraic geometry (the study of solutions of polynomial equations in several variables). Algebraic geometry is fundamental to many applications of mathematics -- such as those to physics, computer science and, more recently, biology -- and was already studied in the work of the ancient Greeks, if not earlier. However, the subject experienced a spectacular revival in the recent past, thanks largely to the work of Alexander Grothendieck and his collaborators. One of the major insights gained from this work was that the study of "approximate" solutions (in the sense of modular or "clockwork" arithmetic) to polynomial equations sheds quite a bit of light on the "true" solutions. In this project, the investigator will expand the techniques available to study "approximate" solutions, and contribute to bridging the gap between "approximate" and "true" solutions.
The goal of this project is to study algebraic geometry in the positive characteristic and p-adic settings. First, with Morrow and Scholze, the PI intends to give an algebro-geometric construction of Breuil-Kisin modules (or p-adic shtukas) associated to p-adic manifolds; this may be viewed as a p-adic analogue of the association of a Hodge structure to a complex manifold, and would have new consequences for the cohomology of algebraic varieties. Secondly, with Scholze, the PI plans to study algebraic geometry over perfect schemes, and its relation to the h-topology; the sought-for descent result for vector bundles would be used to construct determinant line bundles for certain complexes of sheaves that are not linear over the structure sheaf, which will give a new source of algebraic cycles. Finally, with Esnault and Kindler, the PI will work on extensions of Gieseker conjecture (relating fundamental groups to D-modules in characteristic p) and the Grothendieck conjecture (relating stratifying cohomology with the perfection of coherent cohomology). The central theme running through all these projects is the systematic use of "large" objects (such as the pro-etale and h- topologies, perfect and derived schemes) to study "small" objects (such as torsion in cohomology, or line bundles on Grassmanians).
