CAREER: New algebraic techniques for line-point incidence problems

关于线的排列问题在各个数学领域一直被广泛研究。尽管它在数学中起着核心作用，但这一领域的一些最基本的问题仍未得到解答。这个研究项目的主要目标是开发研究直线排列的新技术，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新技术，用于在该领域取得重大进展，并将这种理解应用于计算机科学中的问题。研究者还将继续致力于各级学生的教育和指导，为该研究主题领域的新课程开发和传播材料，并组织指导式研讨会，使学生了解主要的研究趋势和新兴技术。这个研究项目主要关注两大类问题。Kakeya类型的问题是关于将指向不同方向的线组合成一个“小”集合的“最佳”方法。这种类型的问题出现在各种环境中，包括分析、偏微分方程、数论、组合学和理论计算机科学。首席研究员引入了一种称为“多项式方法”的新技术来研究这类问题，并利用它给出了Wolff有限域Kakeya猜想的完整解。本项目将继续以各种方式发展多项式方法，并将其用于解决其他问题。在Sylvester-Gallai型问题中，人们希望将点集中的局部依赖信息转换为整个集合维度上的全局边界。首席研究员开发了一种技术，通过限定“设计矩阵”的秩来研究这种形式的问题。本项目将进一步发展这种方法，并利用它在入射几何和加性组合学的几个核心问题上取得进展。

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.