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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学科密切相关。例如，PI研究的聚类代数出现在数学（表示理论、几何、组合学、动力系统）和物理学（弦理论、散射振幅、可积系统）的许多领域。大约在2000年发现的簇代数为许多问题提供了一个惊人的统一框架。PI感兴趣的是加深这些联系，并找到数学和物理合二为一的更多方法。这些项目也为研究生提供了研究训练的机会。PI将从事的四个项目涉及聚类代数、Ising模型和Kazhdan-Lusztig理论。在第一个项目中，PI计划继续他对周期和可积t系统进行分类的工作，并研究它们的性质。在第二个项目中，PI计划用张量图的方法来研究更高的Teichmuller理论。在最近的一项工作中，PI和一位合著者使用了Postnikov的平面图，从不等式的角度对Ising模型的相关函数进行了全面的描述。在第三个项目中，PI计划利用这些结果来推导相关函数极限的性质，并将这些结果扩展到其他统计力学模型中。最后，在第四个项目中，PI计划使用他开发的广义Robinson-Schensted插入来研究仿射Kazhdan-Lusztig理论。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。