Collaborative Research: Derived Categories in Birational Geometry, Enumerative Geometry, and Non-commutative Algebra

Solutions to polynomial equations form geometric shapes (for example lines, planes, curves). Thus, we can solve equations using geometry (by finding where lines, planes, curves intersect). These ideas have evolved into algebraic geometry, a field of modern mathematics with applications ranging from physics to cybersecurity. In this collaborative project we are particularly interested in the connections to physics. These connections are made through “derived categories”, a complex system of mathematical data associated to a geometric shape. While derived categories have experienced explosive development in recent decades, many questions about them remain unsolved. This project attempts to solve some of the central and most recent questions about derived categories using techniques developed by the principal investigators and many others over the last decade. This award will also support undergraduate and graduate students. This project focuses on three specific questions about derived categories. Together, these questions incorporate numerous areas of mathematics including birational geometry, enumerative geometry, and non-commutative algebra. Specifically, our first question studies connections between derived categories and birational geometry (and the possibly concealed connection between flips/flops and derived partial compactifications). We ask whether K-equivalent varieties have equivalent derived categories, a central conjecture about derived categories due to Bondal-Orlov and Kawamata. Our second question asks how decompositions of quantum cohomology are related to semi-orthogonal decompositions of derived categories. This connection is conjecturally made through mirror symmetry as proposed by Kontsevich and Kuznetsov. Finally, our third question asks how derived categories shed light on resolutions of singularities (namely as moduli spaces coming from non-commutative algebra). Here, we aim to construct non-commutative crepant resolutions using ideas from homological mirror symmetry. The existence of these resolutions has been conjectured by Van den Bergh. As a whole, this project aims to interpolate between these three questions/conjectures using techniques from geometric invariant theory, non-commutative algebra, derived algebraic geometry, and mirror symmetry. The central theme is the explicit use and construction of Fourier-Mukai kernels whose geometry provides a foothold into understanding these problems. Combining the past work of the principal investigators on the construction of Fourier-Mukai kernels, wall crossing for derived categories, and virtual fundamental cycles, we expect to advance our understanding of these fundamental questions. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.