CAREER: New algebraic techniques for line-point incidence problems

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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.