Derived Torelli Theorems, Brauer Degeneration and Universality, and Foundations of Algebraic Vision

University Of Washington, Seattle WA

Investigators

Abstract

This award supports research on several projects in algebraic geometry and allied fields. Focused primarily on solving polynomial equations, algebraic geometry is an ancient subject that plays a key role in numerous fields of mathematics, both pure and applied. It is a linchpin of modern number theory, a heavy hammer of contemporary cryptography, and a crucial input into the computer vision systems that are transforming geography, archaeology, medicine, the automotive industry, and consumer smartphones. While it continues to expand its abstract foundations in astounding new directions, algebraic geometry is also cutting new paths into data science, statistics, and machine learning. The questions under study in this project attack both ends of this spectrum. One part of the project will focus on problems related to so-called Torelli theorems, which seek to capture and quantify the essential linearity of algebro-geometric objects, and on the Brauer group, which is an object that tightly binds algebraic geometry to mathematical physics, non-commutative algebra, and number theory. A second part of the project aims to broaden the algebro-geometric foundations of computer vision, bringing new approaches to deep problems that lie at the heart of cutting-edge applications of computer vision. The first part of this research project concerns Torelli theorems and focuses on Torelli-type statements for various mixtures of derived categories and Chow theory, building on the investigator's earlier work with collaborators. This is one way of trying to get Torelli theorems in a positive characteristic setting, and the underlying ideas have already paid dividends related to the Tate conjecture for surfaces over finite fields. Research on the Brauer group aims at new degeneration methods via semistable reduction of maximal orders and universality questions for rational Brauer groups of projective spaces. The goal is to gain structural insight into Brauer classes and attack several old problems, such as the cyclicity conjecture. The part of the project on computer vision will introduce functorial methods into the study of multiview reconstruction and resection, leading to new ways of compactifying the natural incidence correspondences that occur in these problems. The work aims to help lay new flexible foundations for algebraic geometry in computer vision that will advance the relationship between the subjects.