CAREER: Analytic and Spectral Methods in Combinatorics

这个职业奖支持组合学的研究。PI旨在开发和应用新的分析和谱方法，用于离散数学中的各种极值问题。例如，这样的调查会问一个（点、向量、整数等）集合有多大。满足一定的组合性质。结果在数论、计算机科学和其他领域都有应用。分析和谱方法已经在该领域发挥了核心作用;该项目将导致新技术在组合学及其他领域的进一步应用。该项目还将有一个教育部分，包括研究生导师和课程开发。研究方向之一涉及本征值多重性和应用于等角线，球面双距离集，以及更普遍的高维球面码。PI最近通过谱图理论的新见解解决了等角线上的一个长期问题，为通过更广义的谱图理论问题进一步发展高维球形码铺平了道路。第二个方向涉及图论和添加剂组合学中的极值问题，专注于连接这两个领域的技术。PI将研究稀疏正则性方法、图同态密度不等式（例如Sidorenko猜想）、超图中的极值问题以及相关主题。该奖项反映了NSF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。

This CAREER award supports research in combinatorics. The PI aims to develop and apply novel analytic and spectral methods designed for a variety of extremal problems in discrete mathematics. Such investigations ask, for example, how large a set (of points, vectors, integers, etc.) satisfying certain combinatorial properties can be. Results have applications in number theory, computer science, and other fields. Analytic and spectral methods already play central roles in the field; this project will lead to new techniques with further applications in combinatorics and beyond. The project also will have an educational component, including graduate student mentorship and course development.One of the directions of research concerns eigenvalue multiplicities and applications to equiangular lines, spherical two-distance sets, and more generally spherical codes in high dimensions. The PI recently solved a longstanding problem on equiangular lines via new insights in spectral graph theory, paving the way for further developments on high dimensional spherical codes via more generalized spectral graph theoretic problems. A second direction concerns extremal problems in graph theory and additive combinatorics, focusing on techniques connecting the two areas. The PI will study sparse regularity methods, graph homomorphism density inequalities (for example Sidorenko’s conjecture), extremal problems in hypergraphs, and related topics.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.