Combinatorics and Braid Varieties

组合学和编织品种

• 批准号: 2246877
• 负责人: Nathan Williams
• 金额: $ 21万
• 依托单位: University of Texas at Dallas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246877&HistoricalAwards=false
• 关键词: Combinatorics; Braid; Varieties

Algebraic combinatorics is a branch of mathematics that studies algebraic structures using combinatorial methods, and combinatorial structures using algebraic methods. Such structures and methods are often fundamental to many different scientific disciplines and show up in many different contexts---algebraic combinatorics therefore has diverse applications to subjects like cryptography, protein folding, high-energy physics, and quantum computing. This project will use algebraic objects and methods to produce new combinatorial results, leveraging braid varieties--a sort of algebraic space associated to a knot--as a unifying tool. Funds will additionally support training graduate students and outreach efforts, including work on an interactive online discrete mathematics textbook.In more detail, this proposal suggests a framework for producing combinatorial results using braid varieties over finite fields, Hecke algebra traces, rational Cherednik algebras, and a new relationship with noncrossing combinatorics. The framework has already proven successful in producing substantial new results: the PI's recent joint work with Galashin, Lam, and Trinh resolved two decades-long open problems in Coxeter-Catalan combinatorics, simultaneously producing the first definition of rational noncrossing Coxeter-Catalan objects, while also giving the first uniform enumeration of noncrossing objects. Connections to Macdonald theory--diagonal harmonics and q,t-combinatorics--are also expected when working over the complex numbers. At different levels of generality, different techniques become available. For finite Coxeter groups, it is possible to compute everything in a case-by-case manner using an explicit decomposition of the Hecke algebra, and there are many interesting combinatorial and representation-theoretic problems open for immediate attack. Special classes of elements in finite type have favorable representation-theoretic properties that allow for uniform approaches. For affine Weyl groups, the main tool is a trace formula for translations, due to Opdam. For example, the proposed framework recovers some Tessler matrix identities due to Haglund in this setting. For general Kac-Moody Weyl groups, we are reduced to general recursive and cluster-theoretic methods. These methods also apply in both the finite and affine cases, but will require software implementation before further exploration is possible.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.

代数组合学是数学的一个分支，它用组合方法研究代数结构，用代数方法研究组合结构。这种结构和方法通常是许多不同科学学科的基础，并在许多不同的背景下出现-代数组合学因此在密码学、蛋白质折叠、高能物理和量子计算等学科中有不同的应用。这个项目将使用代数对象和方法来产生新的组合结果，利用辫子簇--一种与纽结相关的代数空间--作为统一工具。此外，基金还将支持研究生的培训和推广工作，包括交互式在线离散数学教科书的工作。更详细地，这项建议提出了一个框架，用于产生组合结果，使用有限域上的辫子簇、Hecke代数迹、有理Cherednik代数，以及与非交叉组合数学的新关系。该框架已经被证明在产生实质性的新结果方面是成功的：PI最近与Galashin、Lam和Trin的合作解决了Coxeter-Catalan组合学中长达20年的公开问题，同时给出了有理非交叉Coxeter-Catalan对象的第一个定义，同时也给出了第一个非交叉对象的统一枚举。与麦克唐纳理论的联系--对角调和和q，t-组合学--在处理复数时也是可以预料到的。在不同的一般性级别上，可以使用不同的技术。对于有限Coxeter群，可以使用Hecke代数的显式分解以逐个情况的方式计算一切，并且有许多有趣的组合和表示论问题可以立即攻击。有限类型的特殊元素类具有良好的表示理论性质，允许采用统一的方法。对于仿射Weyl群，主要的工具是一个用于平移的迹公式，这是由于Opdam。例如，在这种情况下，所提出的框架恢复了由于Haglund的一些Tessler矩阵恒等式。对于一般的Kac-Moody Weyl群，我们归结为一般的递归方法和簇论方法。这些方法也适用于有限和仿射情况，但在可能进行进一步探索之前，需要软件实现。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。