Reconstruction Theorems, Brauer Groups, and Algebraic Vision

代数几何是研究可以用多项式方程定义的形状的学科。它具有巨大的表达能力，导致其纳入许多应用和纯数学学科。这是一个古老的课题，几个世纪以来经历了几次大的变迁。最近数学上的许多突破都依赖于代数几何技术。在其经典形式中，代数几何在摄影测量和计算机视觉中发挥着重要作用，产生了每天在世界各地运行数十亿次的关键计算技术和算法。在其现代形式中，它是密码学，互联网通信和国家安全的基石。它是现代数论中一个重要的技术工具。PI将研究该主题的几个领域的问题。他将继续他的工作，衍生类别和重建结果，侧重于过滤衍生等价物的性质和新的拓扑重建技术，类似于经典的结果Torelli，加布里埃尔，和其他人，已开发的PI和他的合作者。他将扩大研究的布劳尔组从品种刚性分析空间，旨在建立基础成果刚性空间，扩展已知结果的经典分析空间。他还将继续使用现代技术研究计算机视觉的代数几何，建立在PI和合作者近年来所做的工作基础上。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.