Cluster Algebras, Atomic Bases, and String Theory

在数学中，一个公式通常可以从多个角度产生。例如，高中代数中的二次公式用于找到多项式方程的根，但也可以几何地指示直线和抛物线相交的位置。 在对所谓的簇代数的研究中，公式通过一个复杂但明确的过程产生，称为簇突变。 在以前的研究中，PI开发了更简单的模型，提供了一种计算这些公式的替代方法，避免了较长的递归过程。 本研究项目将这些模型与弦理论和几何学中出现的模型进行比较，揭示出所有这些课题在此过程中的新特征，并在物理学家、数学家和学生之间创造出令人兴奋的合作。本研究项目探讨了Fomin和Zelevinsky在2001年定义的簇代数在其他数学和物理领域的各种应用。 簇代数是一类由二项式交换关系定义的交换代数，它是为了研究Lusztig的对偶标准基而引入的。 从那时起，许多不同主题的应用程序已经被发现，包括范畴论，代数表示，Teichmüller理论，离散可积系统，多面体组合学和统计物理，仅举几例。 这个研究项目将进一步研究双曲几何和弦理论的联系。 特别是，PI将研究散射图和膜平铺及其与集群代数和集群变量的联系。

Oftentimes in mathematics, a formula can arise from multiple perspectives. For example, the quadratic formula from high school algebra is used to find the roots of a polynomial equation but also indicates geometrically where a line and a parabola intersect. In research on so-called cluster algebras, formulas arise via a complicated but explicit process known as cluster mutation. In previous research, the PI developed simpler models that give an alternative way to compute these formulas and avoid the longer recursive procedure. This research project compares these models to those arising in string theory and geometry, revealing new features of all of these topics in the process, and creating exciting collaborations among physicists, mathematicians, and students.This project investigates various applications of cluster algebras, defined by Fomin and Zelevinsky in 2001, to topics in other fields of mathematics and physics. Cluster algebras are certain commutative algebras defined by binomial exchange relations introduced to study Lusztig's dual canonical bases. Since then, applications to many different topics have been discovered, including category theory, quiver representations, Teichmüller theory, discrete integrable systems, polyhedral combinatorics, and statistical physics, just to name a few. This research project will further investigate the connections to hyperbolic geometry and string theory. In particular, the PI will investigate scattering diagrams and brane tilings and their links to cluster algebras and cluster variables.