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

PI对代数组合学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互动。例如，PI研究的簇代数出现在数学(表示论、几何、组合学、动力系统)和物理(弦理论、散射振幅、可积系统)的许多领域。大约在2000年发现的星系团代数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，并找到更多使数学和物理成为一门科学的方法。这些项目还为研究生提供了研究培训的机会。PI将致力于的四个项目涉及簇代数、伊辛模型和Kazhdan-Lusztig理论。在第一个项目中，PI计划继续他的工作，对周期和可积T-系统进行分类，以及研究它们的性质。在第二个项目中，PI计划用张量图的方法来研究更高的泰希穆勒理论。在最近的一项工作中，PI和一位合著者使用波斯尼科夫的可变图，以不等式的形式给出了伊辛模型的关联函数的全面描述。在第三个项目中，PI计划使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。最后，在第四个项目中，PI计划使用他开发的广义Robinson-Schensted插入法来研究仿射Kazhdan-Lusztig理论。该奖项反映了NSF的法定使命，并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。

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.