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; 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，使用组合方法研究代数结构，并使用代数方法进行组合结构。 这种结构和方法通常是许多不同科学学科的基础，并在许多不同的情况下出现 - 因此，代数组合主义者对密码学，蛋白质折叠，高能物理学和量子计算等受试者具有不同的应用。 该项目将使用代数对象和方法来产生新的组合结果，利用编织品种（与结相关的代数空间）作为统一工具。 资金还将支持培训研究生和外展工作，包括在互动的在线离散数学教科书上的工作，此建议提出了一个框架，该框架是使用有限田的编织品种，Hecke代数痕迹，Rational Cherednik代数和与非交叉组合的新框架一起产生组合结果的框架。 该框架已经成功地产生了实质性的新结果：PI最近与Galashin，Lam和Trinh的联合合作解决了Coxeter-Catalan Combinatorics的长达二十年的开放问题，同时生成了合理的非交叉Coxeter Coxeter-catalan对象的第一个定义，同时也提供了第一个不合格的非涂抹对象。 与麦克唐纳理论的连接 - 二角形谐波和Q，T-Combinatorics - 在处理复数时也可以预期。 在不同级别的一般性下，可以使用不同的技术。 对于有限的Coxeter组，可以使用Hecke代数的明确分解以逐案方式计算所有内容，并且有许多有趣的组合和表示理论问题，以立即攻击。 有限类型中的特殊类别的元素具有有利的表示理论特性，可以实现统一的方法。 对于仿射韦尔组，主要工具是由于OPDAM而导致的痕量公式。 例如，所提出的框架在此设置中由于haglund而恢复了一些Tessler矩阵身份。 对于一般的Kac-Moody Weyl群，我们被简化为一般递归和群集理论方法。 这些方法也适用于有限案例和仿射案例，但需要在进行进一步探索之前进行软件实施。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获得支持。