Algebra and Combinatorics of Free Structures

自由结构的代数和组合学

• 批准号: 0600973
• 负责人: Marcelo Aguiar
• 金额: $ 10.97万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-15 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600973&HistoricalAwards=false
• 关键词: Algebra; Combinatorics; Free; Structures

This project will deal with a variety of algebraic objects that havearisen recently in different areas such as Hopf algebra theory,combinatorial representation theory,the theory of operads, free probability theory, and renormalization theoryin mathematical physics. It should be regarded as an extension of the classical work on the theory of free Lie algebras and its interplay with the representation theory of the symmetric group.Through the eyes of operad theory, the objects we deal with are free algebras of various kinds. Addressing this property is fundamental in understanding the roleplayed by these objects in the other theories.Freeness is also responsible for the rich combinatorics exhibited by theseobjects. Part of the goal of the project is to make the algebraic structure as explicit as possible, which often leads to interesting combinatorial constructions.Conversely, the project will make use of algebraic properties to unify and generalize important constructions in combinatorics.Some of these free algebras have foundapplications in free probability theory, while others are at the basis ofrenormalization theory. All of them are closely related to symmetric functionsand the representation theory of the symmetric group. A common feature is theexistence of Hopf algebraic structures, which are often also free in one senseor another. Exploiting this structure is another central feature of the project.Combinatorics provides a tool for handling subtle algebraic structures.Understanding these structures is important in various areas of recent interestboth in pure mathematics (free probability theory, representation theory) as well as in mathematical physics (renormalization theory).This project plans to deepen our understanding of the rich combinatoricsunderlying various free algebraic structures.The PI will introduce graduate students to this area of research and makeinternational contacts and collaborations in Europe, Latin America, and India.

这个项目将处理最近在不同领域的各种代数对象，如Hopf代数理论、组合表示理论、算术理论、自由概率理论和数学物理中的重整化理论。它应该被看作是自由李代数理论的经典工作及其与对称群的表示理论的相互作用的扩展。解决这一性质是理解这些对象在其他理论中所扮演的角色的基础。自由性也是这些对象所展示的丰富的组合学的原因。该项目的部分目标是使代数结构尽可能明确，这往往会导致有趣的组合结构。相反，该项目将利用代数性质来统一和推广组合中的重要结构。这些自由代数中的一些在自由概率论中有广泛的应用，而另一些则基于重整化理论。它们都与对称函数和对称群的表示理论密切相关。一个共同的特征是存在Hopf代数结构，这些结构通常在某种意义上也是自由的。利用这种结构是这个项目的另一个主要特点。组合学为处理微妙的代数结构提供了一个工具。了解这些结构在最近的各个领域都很重要，无论是在纯数学(自由概率论，表示论)还是在数学物理(重整化理论)中。这个项目计划加深我们对各种自由代数结构背后丰富的组合学的理解。PI将向研究生介绍这一研究领域，并在欧洲、拉丁美洲和印度进行国际联系和合作。