FRG: Collaborative Research: Higher Categorical Structures in Algebraic Geometry

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project aims to apply major new developments in mathematics to open questions in algebra and algebraic geometry. Algebra is the study of generalized systems of numbers, while algebraic geometry is concerned with the geometry of solutions of polynomial equations. Both fields are used throughout mathematics and touch regularly on daily life via algorithms used in computer vision (for instance in cell phone cameras), satellite communications (error-correcting codes), and secure messaging (cryptography using elliptic curves). The project also uses higher category theory developed over the last two decades, which makes it possible to systematically deal with subtle, loosely defined objects. This extra flexibility leads to new control over the basic objects used in algebraic geometry. Even more recently, some work on condensed mathematics raises the possibility of extending this new control to closely related areas of analysis. This project will use this cutting-edge work to attempt to settle longstanding questions in algebraic geometry and to introduce and solve new questions in analytical algebraic geometry. It will provide research and training opportunities for graduate students and postdoctoral researchers and will support several workshops aimed at early-career mathematicians. There are four main research challenges addressed by this project. First, the PIs aim to find complete noncommutative categorical invariants and to find a bridge directly from the topological invariants to the categorical ones. No known noncommutative categorical invariant suffices to reconstruct an algebraic variety. In good cases, work of the PIs and collaborators shows that the underlying space is enough for such a reconstruction. Next, to clarify the role of commutative objects inside noncommutative objects, the PIs will study the deformations and local systems of dg categories in an attempt to settle Orlov's geometricity conjecture. Third, the PIs will study the p-adic cohomology of algebraic varieties via higher categorical invariants such as topological Hochschild homology, applied to the derived category. Finally, the PIs will try to show that the recently constructed theory of nuclear modules yields the correct noncommutative invariants of a rigid analytic variety and will aim to generalize the first three projects to the more general analytic context. 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.