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

该奖项支持在代数几何和相关领域的几个项目的研究。代数几何主要关注多项式方程的求解，是一门古老的学科，在数学的许多领域，无论是纯粹的还是应用的，都发挥着关键作用。它是现代数论的关键，是当代密码学的重锤，也是计算机视觉系统的关键输入，而计算机视觉系统正在改变地理学、考古学、医学、汽车工业和消费者智能手机。当代数几何继续在惊人的新方向上扩展其抽象基础时，它也在数据科学、统计学和机器学习方面开辟了新的道路。在这个项目中研究的问题攻击了这一范围的两端。该项目的一部分将集中于所谓的托雷利定理（Torelli theorem）和布劳尔群（Brauer group），前者旨在捕捉和量化代数几何对象的基本线性，后者是将代数几何与数学物理、非交换代数和数论紧密联系在一起的对象。该项目的第二部分旨在拓宽计算机视觉的代数-几何基础，为计算机视觉前沿应用的核心问题带来新的方法。该研究项目的第一部分涉及Torelli定理，并以研究者早期与合作者的工作为基础，重点关注派生类别和Chow理论的各种混合的Torelli型陈述。这是一种尝试在正特征设置中得到托雷利定理的方法，其基本思想已经与有限域上曲面的泰特猜想相关。Brauer群的研究旨在通过极大阶的半稳定约简得到新的退化方法和射影空间的有理Brauer群的普适问题。我们的目标是深入了解Brauer类的结构，并解决几个老问题，比如循环猜想。计算机视觉部分将把函数方法引入到多视图重建和切除的研究中，从而找到新的方法来紧化这些问题中出现的自然关联对应。这项工作旨在帮助为计算机视觉中的代数几何奠定新的灵活基础，这将促进学科之间的关系。

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.