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

导出的托雷利定理、布劳尔退化和普遍性以及代数视觉基础

• 批准号: 1600813
• 负责人: Max Lieblich
• 金额: $ 31.5万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600813&HistoricalAwards=false
• 关键词: Derived; Torelli; Theorems; Brauer; Degeneration

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.

该奖项支持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求解多项式方程，是一门古老的学科，在数学的许多领域中发挥着关键作用，无论是纯数学还是应用数学。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，这些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惊人的新方向上扩展其抽象基础，但它也在数据科学，统计学和机器学习方面开辟了新的道路。本项目所研究的问题涉及这一范围的两端。该项目的一部分将集中在与所谓的Torelli定理相关的问题上，该定理试图捕捉和量化代数几何对象的基本线性，以及Brauer群，这是一个将代数几何与数学物理，非交换代数和数论紧密结合在一起的对象。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，重点关注衍生类别和Chow理论的各种混合的Torelli类型陈述，建立在研究者与合作者的早期工作基础上。这是试图在正特征背景下获得Torelli定理的一种方法，并且其基本思想已经为有限域上的曲面带来了与泰特猜想相关的红利。Brauer群的研究旨在通过投影空间有理Brauer群的最大阶的半稳定约化和普遍性问题来寻求新的退化方法。目标是获得对Brauer类的结构洞察，并解决几个老问题，如循环猜想。计算机视觉项目的一部分将把函子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压缩这些问题中出现的自然关联对应。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，从而促进学科之间的关系。