Geometry in combinatorics

这个项目是关于组合学和几何学与其他数学的联系。PI计划调查的主要现象有三个。第一个是刚性几何结构的出现，作为图兰和拉姆齐型问题的答案（或简化的答案）。在这里，我们的目标是理解为什么这些结构会出现，并找到一种系统的方法从组合问题到这样的结构。第二个现象出现在大点集的近似问题中。在这里，问题的两个方面导致两个不同的领域：代数拓扑的下界和逻辑的上界。PI将致力于扩展现有的技术，目标是完善它们，以消除目前需要的猜测。最后一个现象涉及几何关联问题。PI将尝试将现有的拓扑方法推广到有限域设置。这里的动机是在欧几里得空间中的证明之间的表面相似性，如不同距离问题的解决，和有限域设置中稠密集的和积型结果的证明。这个项目的目标是加强组合数学与数学领域的联系，这些领域通常与组合数学无关。 除了在具体的组合问题上取得进展的直接好处外，目的是建立桥梁，使不同领域的专家能够分享见解。此外，几何能够直接吸引我们的想象力，因此可以作为组合教育的催化剂。

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.