Derived Categories and Other Invariants of Algebraic Varieties

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The project is focused on a variety of questions in algebraic geometry, loosely centered around the study of invariants, both cohomological and categorical. Algebraic geometry is concerned with the study of the geometric spaces which locally can be modeled as the set of solutions of systems of polynomials in several variables, typically over a field but also over arithmetically interesting rings such as the integers. Such spaces are ubiquitous in mathematics, arising in many other areas such as representation theory, combinatorics and mathematical physics. They also arise naturally in a variety of applications in other disciplines such as computer science and engineering. A common theme in mathematics when studying complex objects, such as algebraic varieties, is to consider various features or invariants which are more tractable yet rich enough to capture meaningful information. The project will advance our understanding of several such invariants and study applications in algebraic and arithmetic geometry.More specifically the project is concerned with the following topics. Most of the work will be carried out with collaborators and students. (1) Derived categories of coherent sheaves on smooth projective algebraic varieties over a field and understanding additional cohomological features of equivalences between such categories. The goal is understand to what extent the derived category of coherent sheaves, together with additional structure, determines the isomorphism class, or possibly the birational equivalence class, of an algebraic variety. (2) Log coherent sheaves and Hochschild and topological Hochschild homology for log schemes. The objective is to understand a suitable dg category of coherent sheaves associated to a morphism of log schemes. The investigation of such a category also has broader implications for the study of degeneration phenomena for coherent sheaves. (3) Reconstruction theorems for algebraic varieties. In particular, the PI will investigate new approaches to, and generalizations of, reconstruction results previously studied using model theory. (4) Homotopy groups of log schemes and applications to the study of crystalline fundamental groups. In addition to these four projects, the PI will investigate several other projects together with graduate students.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.