Algebraic and Probabilistic Methods in Extremal Combinatorics

极值组合中的代数和概率方法

基本信息

批准号：
2100157
负责人：
Lisa Sauermann
金额：
$ 11.09万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100157&HistoricalAwards=false
关键词：
Algebraic Probabilistic Methods Extremal Combinatorics

项目摘要

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.

极端组合学是数学领域，它研究了在某些约束下可以在数学对象的大小或小配置。该领域正在迅速发展，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。该研究项目旨在使用代数和概率理论的方法在极端组合学中的问题上取得进展。该研究的重点是一些长期存在的开放问题和猜想，以及一些相关的问题。这项工作将导致开发新的数学工具和技术，并突破已知方法的限制。此外，通过她的教学和指导，研究人员努力鼓励学生了解数学并从事STEM领域的职业。该项目中研究的问题属于两个粗略的主题领域。这些领域中的第一个集中在2016年引入的切片等级方法上。该方法导致了添加剂组合学的几个壮观的结果，但许多相关的问题仍然开放。研究人员打算研究特定问题，以说明切片等级方法的当前局限性。该项目的目的之一是找到使该方法更灵活且更广泛地适用的方法。该项目的第二个主题领域涉及诱导性问题，该问题是在四十年前提出的，并且仍然很开放。给定固定的图H和一个大整数N，此问题询问了N-Vertex图可以包含的最大诱导副本数。该区域的一个主要开放问题是图H是路径或周期的情况。使用概率技术，PI计划调查这个问题以及其他诱导性类型问题。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来提供支持。