Algebraic Methods in Extremal Combinatorics

极值组合学中的问题要求在一组约束条件下优化一个组合不变量。这个刻意广泛的问题在工业和整个数学中都有无处不在的应用，但也导致了一些本身就很有趣的问题。这个项目将尝试对以下形式的陈述进行量化：如果一个对象在适当的意义上是“大的”，那么它包含有序的子结构。该项目将特别寻求使用代数和几何的方法来回答极值图论和组合数论中的问题。这个项目的主要好处有两个：第一，它将回答几个数学领域交叉的重要开放问题，第二，它将涉及本科生的研究。PI将特别关注Turan和Ramsey类型的问题，加法组合学，以及谱图理论。代数和几何结构对于极值图论问题一直很有用，这个项目将继续使用这些方法，并探索它们的局限性和相互之间的联系。该项目还将寻求将这两个领域的已知方法扩展到超图，因为我们的知识有限。该项目的目标位于组合学、代数、组合数论和几何的交叉点上，因此许多技术都适用，来自几个不同领域的问题将被解决。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Questions in extremal combinatorics ask to optimize a combinatorial invariant subject to a set of constraints. This deliberately broad question has ubiquitous applications both in industry and throughout mathematics, but also leads to questions that are interesting in their own right. This project will attempt to quantify statements of the form: if an object is "large" in a suitable sense, then it contains ordered substructure. The project will specifically seek to use methods from algebra and geometry to answer problems in extremal graph theory and combinatorial number theory. The main benefits of the project are two-fold: first it will answer important open problems in the intersection of several mathematical areas, and second it will involve undergraduate students in research.The PI will focus specifically on Turan and Ramsey type problems, on additive combinatorics, and on spectral graph theory. Algebraic and geometric constructions have long been useful for extremal graph theory problems, and this project will continue to use these methods as well as explore their limitations and connections to each other. The project will also seek to extend known methods in both areas to hypergraphs, where our knowledge is limited. The project goals sit in the intersection of combinatorics, algebra, combinatorial number theory, and geometry, and as such many techniques are applicable and problems from several different areas will be worked on.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.