Reconstruction Theorems, Brauer Groups, and Algebraic Vision

University Of Washington, Seattle WA

Investigators

Abstract

Algebraic geometry is the study of shapes that can be defined using polynomial equations. It has immense expressive power, leading to its incorporation into numerous applied and pure mathematical disciplines. It is an ancient subject, experiencing several flowerings over centuries. Many recent breakthroughs in mathematics rely on algebro-geometric techniques. In its classical form, algebraic geometry plays an essential role in photogrammetry and computer vision, yielding key computational techniques and algorithms that are run billions of times per day all over the world. In its modern form, it is a cornerstone of cryptography, internet communications, and national security. It is a crucial technical tool in modern number theory. The PI will study problems in several areas of the subject. He will continue his work on derived categories and reconstruction results, focusing on the properties of filtered derived equivalences and on new topological reconstruction techniques, analogous to classical results of Torelli, Gabriel, and others, that have been developed by the PI and his collaborators. He will expand the study of the Brauer group from varieties to rigid analytic spaces, aiming to establish foundational results for rigid spaces that extend known results for classical analytic spaces. He will also continue his study of the algebraic geometry of computer vision using modern techniques, building on work done by the PI and collaborators in recent years. This has potential uses in the development of new algorithms. 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.