Problems in Extremal and Geometric Combinatorics

极值和几何组合学的发展是由数学、计算机科学和其他学科的许多领域的问题所影响的。该项目的目的是开发广泛适用的通用组合工具和解决具体的组合问题。该项目将对研究生和本科生进行培训。要解决的具体问题包括与图兰问题和拉姆齐理论有关的代数和几何问题。我们将特别注意代数构造。这里的目的是了解该地区许多已知建筑的明显刚性，并开发更通用的技术。预计这将导致在有限几何和极值组合之间建立更紧密的联系，丰富这两个领域。该项目还考虑了与单词和子序列相关的几个主题，以及差异理论。潜在的影响包括更好的纠错代码和数值算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The development of extremal and geometric combinatorics has been informed by questions in a number of areas of mathematics, computer science and other disciplines. The aim of the project is both to develop general combinatorial tools of wide applicability and to solve specific combinatorial problems. Graduate and undergraduate students will be trained during this project.The specific problems to be tackled include algebraic and geometric questions related to Turan problems and Ramsey theory. Particular attention will be devoted to algebraic constructions. The aim here is to understand the apparent rigidity of many known constructions in the area, and develop more general techniques. It is expected that this will result in forging the connections between finite geometry and extremal combinatorics closer, enriching both areas. The project also considers several topics related to words and subsequences, as well as discrepancy theory. The potential impact includes better error-correcting codes and numerical algorithms.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.