Algebraic and Probabilistic Methods in Extremal Combinatorics

极值组合是研究数学对象在特定约束条件下的大小配置的数学领域。这个领域正在迅速发展，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问题上取得进展。研究的重点是一些长期开放的问题和猜想，以及几个相关的问题。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，研究者努力鼓励学生学习数学，并在STEM领域从事职业。本项目研究的问题大致分为两个主题领域。其中第一个领域围绕着2016年引入的切片秩多项式方法。这种方法已经在加性组合学中得到了一些惊人的结果，但是许多相关的问题仍然没有解决。研究者打算研究具体的问题，举例说明当前切片秩多项式方法的局限性。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可诱导性问题，这个问题在四十多年前就提出了，至今仍未解决。给定一个固定的图H和一个大的整数n，这个问题问的是一个n顶点图可以包含的H的最大诱导拷贝数。这个领域的一个主要开放问题是图H是路径还是循环的情况。利用概率技术，PI计划调查这个问题以及其他可归纳性类型的问题。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。

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.