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。代数几何主要专注于解决多项式方程，是一门古老的学科，在许多数学领域发挥着关键作用，包括纯数学和应用数学。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，计算机视觉系统正在改变地理学、考古学、医学、汽车工业和消费智能手机。虽然代数几何继续在令人震惊的新方向上扩展其抽象基础，但它也在数据科学、统计学和机器学习中开辟新的道路。这个项目中正在研究的问题触及了这个光谱的两端。该项目的一部分将集中在与所谓的Torelli定理有关的问题上，该定理试图捕捉和量化代数几何对象的本质线性，以及Brauer群，它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，为深入解决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，并基于研究者早期与合作者的工作，重点研究派生范畴和Chow理论的各种混合的Torelli型语句。这是试图在正特征设置下得到Torelli定理的一种方法，其基本思想已经与有限域上的曲面的泰特猜想有关。对Brauer群的研究旨在通过极大阶的半稳定降阶和射影空间的有理Brauer群的普适性问题来寻找新的退化方法。我们的目标是获得对布劳尔类的结构性洞察，并解决几个老问题，例如循环性猜想。计算机视觉部分将把函数论方法引入到多视点重建和切除的研究中，从而产生压缩这些问题中自然关联关系的新方法。这项工作旨在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.