Algebra, combinatorics and mathematical computer science

代数、组合学和数学计算机科学

• 批准号: CRC-2021-00120
• 负责人: Thomas, Hugh
• 金额: $ 10.93万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Canada Research Chairs
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=750789
• 关键词: Algebra; combinatorics; mathematical; computer; science

The scattering amplitudes problem in physics is the problem of describing what happens when elementary particles scatter off each other. As well as being of fundamental theoretical importance, this problem is also of practical significance: a precise knowledge of scattering amplitudes is needed in order to analyze data from particle accelerators such as the Large Hadron Collider.The traditional approach to this problem uses a technique called "Feynman diagrams." In order to solve the scattering amplitudes problem, one sums up contributions from the (many) relevant Feynman diagrams. Some ten years ago, a group of physicists around Nima Arkani-Hamed of the Institute for Advanced Study in Princeton initiated a program to recast the solution to the scattering amplitudes problem in a more holistic way. Their approach has, surprisingly, connected the topic up to a number of topics in pure mathematics in which the candidate is an expert, notably cluster algebras and the representation theory of finite dimensional algebras.The candidate's proposal is two-fold: he will address the new and exciting mathematical questions originating from the physicists' approach, and he will collaborate actively with physicists to apply these results to the scattering amplitudes problem.The physicists' approach to the scattering amplitudes problem revolves around the construction of a geometrical object which contains the essential element of the answer encoded within it. For the quantum field theory under discussion, this requires the construction of a series of polyhedra of increasing complexity. The 0-th polyhedron in the series is the associahedron, originally described by James Stasheff in the 1960's, but seeing renewed interest recently because of its connection to cluster algebras. Together with collaborators, the candidate has shown that the next polytope in the series is also connected to a cluster algebra. The candidate proposes to use cluster algebras (and related finite dimensional algebras) to construct all the needed polyhedra, and also analogous non-polyhedral spaces.The program will shed new light on cluster algebras and representation theory of finite dimensional algebras, as well as driving forward research in scattering amplitudes.

物理学中的散射幅度问题是描述基本粒子相互散射时发生的情况的问题。除了具有基础理论重要性外，这个问题也具有实际意义：为了分析大型强子对撞机等粒子加速器的数据，需要精确了解散射振幅。解决这个问题的传统方法使用一种称为“散射振幅”的技术。 “费曼图。”为了解决散射幅度问题，我们总结了（许多）相关费曼图的贡献。大约十年前，普林斯顿高等研究院尼玛·阿卡尼-哈米德周围的一群物理学家发起了一项计划，以更全面的方式重新制定散射幅度问题的解决方案。令人惊讶的是，他们的方法将该主题与候选人是专家的纯数学中的许多主题联系起来，特别是簇代数和有限维代数的表示论。候选人的提案有两个方面：他将解决源自物理学家方法的新的、令人兴奋的数学问题，他将与物理学家积极合作，将这些结果应用于散射幅度问题。物理学家解决散射幅度问题的方法围绕着构建一个几何对象，其中包含编码在其中的答案的基本元素。对于正在讨论的量子场论，这需要构造一系列复杂性不断增加的多面体。该系列中的第 0 个多面体是关联面体，最初由 James Stasheff 在 1960 年代描述，但最近由于它与簇代数的联系而重新引起人们的兴趣。候选人与合作者一起证明了该系列中的下一个多胞体也与簇代数相关。候选人建议使用簇代数（以及相关的有限维代数）来构造所有需要的多面体以及类似的非多面体空间。该程序将为簇代数和有限维代数的表示论以及驱动提供新的思路散射振幅的前瞻性研究。