Cluster Algebras, the Ising Model, and Affine Kazhdan-Lusztig Cells

簇代数、Ising 模型和仿射 Kazhdan-Lusztig 单元

基本信息

批准号：
1949896
负责人：
Pavlo Pylyavskyy
金额：
$ 33万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1949896&HistoricalAwards=false
关键词：
Cluster Algebras Ising Model Affine

项目摘要

The PI is interested in algebraic combinatorics and its relations to other areas of mathematics. Algebraic combinatorics is a unique area in that it interacts closely with a truly wide range of scientific subjects. For example, cluster algebras studied by the PI arise in many areas of mathematics (representation theory, geometry, combinatorics, dynamical systems) and physics (string theory, scattering amplitudes, integrable systems). Discovered around the year 2000 cluster algebras provide an amazing unifying framework for many questions. The PI is interested in deepening these connections, and finding even more ways in which mathematics and physics are a single science. The projects also provide research training opportunities for graduate students.The four projects the PI will work on pertain to cluster algebras, the Ising model, and Kazhdan-Lusztig theory. In the first project the PI plans to continue his work on classifying periodic and integrable T-systems, as well as studying their properties. In the second project the PI plans to work on a tensor diagram approach to higher Teichmuller theory. In a recent work the PI and a coauthor have used Postnikov's plabic graphs to give a comprehensive description of correlation functions of the Ising model in terms of inequalities. In the third project the PI plans to work on using those results to derive properties of limits of correlation functions, as well as on extending those results to other statistical mechanics models. Finally, in the fourth project the PI plans to use the generalized Robinson-Schensted insertion he developed to study affine Kazhdan-Lusztig theory.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.

PI对代数组合及其与其他数学领域的关系感兴趣。代数组合学是一个独特的领域，因为它与真正广泛的科学主题紧密相互作用。例如，由PI研究的群集代数出现在数学的许多领域（表示理论，几何学，组合学，动力学系统）和物理（弦理论，散射幅度，可集成系统）。在2000年群集代数范围内发现，为许多问题提供了一个惊人的统一框架。 PI有兴趣加深这些联系，并找到更多的数学和物理学是一门科学的方式。这些项目还为研究生提供了研究培训机会。PI将在群集代数，Ising模型和Kazhdan-Lusztig理论方面开展的四个项目。在第一个项目中，PI计划继续他的工作，以分类定期和可整合的T系统，并研究其属性。在第二个项目中，PI计划采用张量图的方法来实现高级Teichmuller理论。在最近的一项工作中，PI和合着者使用后尼科夫的Plabic图来全面描述Ising模型的相关函数，以不平等为方面。在第三个项目中，PI计划使用这些结果来得出相关函数极限的属性，并将这些结果扩展到其他统计力学模型。最后，在第四个项目中，PI计划使用他开发的广泛的Robinson-Schensted插入来研究Aggine Kazhdan-Lusztig理论。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准来通过评估来通过评估来支持的。