Cluster Algebras, Combinatorics, and Knot Theory

当簇代数在2002年由Fomin和Zlevinsky提出时，其最初的动机来自于现代代数的一个分支--表示理论。研究模型的对称性往往比直接研究模型更有成效，表象理论在物理和化学以及其他数学领域都有很多应用。簇代数为贯穿整个表示理论的基本模式提供了一个数学框架。令人惊讶的是，这些模式也在其他科学分支中观察到，这些科学分支先验地与表象理论无关。该项目将在一个已经非常活跃的研究领域提供新的研究方向。研究人员将建立和发展簇代数和其他数学领域之间的关系。这些新的连接允许明确的计算结果以及结构的发展。该项目包括新思想的提出以及对长期悬而未决的问题的调查。该项目集中于簇代数及其与组合学、纽结理论、数论和有限维代数表示理论的关系。调查员将进行几项调查。他将应用他最近发现的簇代数和连分式之间的关系，从一个新的角度研究数论中的经典问题，并应用连分式的机制来开发簇代数中的新工具。他还将建立和发展簇代数和纽结理论之间的一种新的联系，为琼斯多项式和其他纽结不变量提供明确的组合公式。此外，他将继续研究由簇代数产生的有限维非交换代数的表示理论。这一奖项反映了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。

When cluster algebras were introduced by Fomin and Zelevinsky in 2002, their original motivation came from representation theory, which is a branch of modern algebra. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in physics and chemistry as well as in other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns which occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This project will provide new directions of research in an already highly active research area. The investigator will establish and develop relations between cluster algebras and other areas of mathematics. These new connections allow explicit computational results as well as structural development. The project contains the instigation of new ideas as well as the investigation of longstanding open questions.The project focuses on cluster algebras and their relation to combinatorics, knot theory, number theory and representation theory of finite-dimensional algebras. The investigator will pursue several investigations. He will apply his recent discovery of a relation between cluster algebras and continued fractions to study classical problems in number theory from a new point of view and to apply the machinery of continued fractions to develop new tools in cluster algebras. He will also establish and develop a new connection between cluster algebras and knot theory providing explicit combinatorial formulas for Jones polynomials and other knot invariants. Furthermore, he will continue his study of the representation theory of finite-dimensional non-commutative algebras that arise from cluster algebras.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.