Algebraic and Probabilistic Methods in Extremal Combinatorics

极值组合学（Extremal Combinatorics）是一个数学领域，研究数学对象在某些约束下的大小配置。这一领域正在迅速发展，并与许多其他数学领域以及理论计算机科学有着密切的联系。本研究项目旨在利用代数和概率论的方法在极值组合学问题上取得进展。研究的重点是一些长期悬而未决的问题和论点，以及一些相关的问题。这项工作将导致新的数学工具和技术的发展，并推动已知方法的极限。此外，通过她的教学和指导，调查员努力鼓励学生学习数学，并追求在STEM领域的职业生涯。在这个项目中研究的问题分为两个粗略的主题领域。其中第一个领域是围绕2016年引入的切片秩多项式方法。这种方法已经在加法组合学中得到了一些引人注目的结果，但许多相关的问题仍然悬而未决。调查人员打算研究具体问题，举例说明目前的限制切片秩多项式方法。该项目的目的之一是找到使该方法更灵活和更广泛适用的方法。该项目的第二个主题领域涉及归纳问题，这是四十多年前提出的，仍然是开放的。给定一个固定的图H，和一个大整数n，这个问题是关于一个n-顶点图可以包含的H的诱导副本的最大数量。在这一领域的一个主要的开放问题是的情况下，图H是一个路径或循环。PI计划使用概率技术来调查这个问题以及其他诱导型问题。该奖项反映了NSF的法定使命，并且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。

Extremal combinatorics is an area of mathematics that investigates how large or small configurations of mathematical objects can be under certain constraints. This area is rapidly developing and has close connections to many other areas of mathematics, as well as to theoretical computer science. This research project aims to make progress on questions in extremal combinatorics using methods from algebra and probability theory. The research focuses on some longstanding open questions and conjectures, as well as several related problems. The work will lead to the development of new mathematical tools and techniques and push the limits of known methods. Moreover, through her teaching and mentoring, the investigator strives to encourage students to learn about mathematics and to pursue careers in STEM fields.The questions studied in this project fall into two rough topic areas. The first of these areas is centered around the slice rank polynomial method that was introduced in 2016. This method has led to several spectacular results in additive combinatorics, but many related questions remain open. The investigator intends to study specific problems exemplifying the current limitations of the slice rank polynomial method. One aim of this project is to find ways to make the method more flexible and more widely applicable. The second topic area of the project concerns the inducibility problem, which was posed over forty years ago and is still wide open. Given a fixed graph H, and a large integer n, this problem asks about the maximum number of induced copies of H that an n-vertex graph can contain. A major open question in this area is the case where the graph H is a path or a cycle. Using probabilistic techniques, the PI plans to investigate this question as well as other inducibility-type problems.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.