This proposal focuses on questions that arise in the study of algebraic varieties and their cohomology, many of which have arithmetic applications. Earlier work of the PI and others has led to substantial recent developments in our understanding of operations on cohomology arising from correspondences, and also to a new suite of problems which the PI will investigate. The PI will study questions of independence of l for actions of correspondences on various cohomology theories (specifically etale, intersection, and crystalline cohomology), continue his work on localized chern classes, and study global questions using trace formulas. These lines of investigation are motivated by the theory of motives, which predicts independence of l results for certain operators on cohomology and suggests that their traces should have geometric significance.In addition, the PI will continue earlier work researching algebraic stacks, abelian varieties, log geometry, Fourier-Mukai transforms, and moduli spaces. The PI will continue to advise Ph.D. students working on projects related to the proposed research.
The proposed work concerns basic geometric structures, such as cohomology, moduli spaces, and stacks, which lie at the core of algebraic geometry and its interactions with other fields, including number theory, representation theory, combinatorics, and practical applications. Roughly algebraic geometry is concerned with the study of geometric properties of algebraic varieties, which are solutions of systems of polynomial equations. Cohomology theories provide some of the most powerful techniques for the study of such varieties. For example, the Lefschetz trace formula can be used to estimate numbers of solutions of polynomials over finite fields, and more generally cohomology groups of algebraic varieties are one of the main sources of Galois representations, which are fundamental objects in modern number theory. The proposal addresses problems on cohomology theories arising in this context. Another fundamental approach to the study of algebraic varieties is through their classification and moduli spaces. The PI's work on moduli spaces and stacks will continue to advance our understanding of key moduli spaces and provide broadly applicable foundational tools.
