The project is about connections of combinatorics and geometry with the rest of mathematics. There are three main phenomena that the PI plans to investigate. The first is the appearance of rigid geometric structures as answers (or conjectured answers) to Turán- and Ramsey-type problems. Here, the goal is to understand why these structures appear, and to find a systematic way to go from a combinatorial problem to such a structure. The second phenomenon arises in the problems of approximation of large point sets. Here, the two sides of the problem lead to two different areas: the lower bounds to algebraic topology, and the upper bounds to logic. The PI will work on extending the existing techniques, with a goal of perfecting them to the point of eliminating the guesswork that they currently require. The final phenomenon concerns geometric incidence problems. The PI will try to generalize the existing topological approaches to finite field setting. The motivation here is the superficial similarity between the proofs in Euclidean space, such as the resolution of the distinct distance problem, and the proofs of sum-product-type results for dense sets in the finite field setting.
The goal of this project is to strengthen the ties of combinatorics with areas of mathematics that are not commonly linked with combinatorics. Besides the immediate benefit of making progress on concrete combinatorial problems, the aim is to create bridges that will allow specialists in different fields to share insights. Furthermore, geometry is able to appeal to our imagination directly, and consequently can serve as a catalyst in combinatorial education.
