CAREER: New algebraic techniques for line-point incidence problems

职业：线点重合问题的新代数技术

• 批准号: 1451191
• 负责人: Zeev Dvir
• 金额: $ 48.14万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-03-15 至 2021-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1451191&HistoricalAwards=false
• 关键词: CAREER; New; algebraic; techniques; line

Questions about arrangements of lines have been studied extensively in various areas of mathematics throughout the ages. Despite its central role in mathematics, some of the most fundamental questions in this area remain unanswered. This research project's main objective is to develop new techniques for studying arrangements of lines and to make progress on longstanding geometric questions. The investigator will continue to develop new techniques that can be used to make significant advances in this area and to apply this understanding to problems in computer science. The investigator will also continue his commitment to the education and mentoring of students at all levels, develop and disseminate materials for a new course in this research topical area, and organize tutorial-style workshops to expose students to major research trends and emerging techniques.This research project focuses on two broad types of problems. Kakeya type problems ask about the "best" possible way to pack lines pointing in different directions into a "small" set. Questions of this type appear in various contexts including in analysis, partial differential equations, number theory, combinatorics, and theoretical computer science. The principal investigator introduced a new technique, called the "polynomial method" to the study of problems of this kind and used it to give a complete solution to the finite field Kakeya conjecture of Wolff. This project will continue developing the polynomial method in various ways and use it to attack other problems. In Sylvester-Gallai type problems, one wishes to convert information about local dependencies in a point set into global bounds on the dimension of the entire set. The principal investigator developed a technique to study questions of this form by bounding the rank of "design-matrices". This project will develop this method further and use it to make progress on several central questions in incidence geometry and additive combinatorics.

关于线的排列的问题在各个数学领域都得到了广泛的研究。尽管它在数学中的核心作用，在这一领域的一些最基本的问题仍然没有答案。该研究项目的主要目标是开发研究线条排列的新技术，并在长期存在的几何问题上取得进展。 研究人员将继续开发可用于在这一领域取得重大进展的新技术，并将这种理解应用于计算机科学中的问题。 研究员还将继续致力于教育和指导各级学生，开发和传播这一研究主题领域的新课程材料，并组织辅导式研讨会，让学生了解主要的研究趋势和新兴技术。Kakeya类型的问题是关于将指向不同方向的线打包成一个“小”集合的“最佳”可能方法。这种类型的问题出现在各种背景下，包括分析，偏微分方程，数论，组合学和理论计算机科学。首席研究员介绍了一种新的技术，所谓的“多项式方法”的研究问题，这类和使用它来给一个完整的解决方案有限域Kakeya猜想的沃尔夫。这个项目将继续以各种方式开发多项式方法，并使用它来解决其他问题。在Sylvester-Gallai型问题中，人们希望将点集中的局部依赖关系的信息转换为整个集合的维度上的全局边界。主要研究者开发了一种技术，通过界定“设计矩阵”的等级来研究这种形式的问题。这个项目将进一步发展这种方法，并使用它在关联几何和加法组合学的几个中心问题上取得进展。