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Cluster Algebras, Combinatorics, and Knot Theory

簇代数、组合学和结理论

基本信息

批准号：
1800860
负责人：
Ralf Schiffler
金额：
$ 15万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-15 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800860&HistoricalAwards=false
关键词：
Cluster Algebras Combinatorics Knot Theory

项目摘要

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.

当簇代数在2002年由Fomin和Zelevinsky引入时，它们的最初动机来自表示论，这是现代代数的一个分支。研究模型的对称性往往比直接研究模型更有成果，表示论在物理和化学以及其他数学领域中有许多应用。簇代数为整个表示论中出现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，先验地，与表征理论无关。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集群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。该项目包含新思想的激发以及长期未决问题的调查。该项目侧重于簇代数及其与组合学，纽结理论，数论和有限维代数的表示理论的关系。调查员将进行几项调查。他将应用他最近发现的一个关系集群代数和继续分数研究经典问题的数论从一个新的角度来看，并适用于机械的继续分数发展新的工具集群代数。他还将建立和发展集群代数和结理论之间的一个新的连接提供明确的组合公式琼斯多项式和其他结不变。此外，他将继续他的研究表示理论的有限维非交换代数所产生的集群代数。这个奖项反映了美国国家科学基金会的法定使命，并已被认为是值得的支持，通过评估使用该基金会的智力价值和更广泛的影响审查标准。