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当基本粒子彼此散射时会发生什么的问题。除了具有基本理论重要性的基本重要性外，此问题还具有实际意义：为了分析来自粒子加速器（例如大型强子对撞机）的数据，还需要精确的散射幅度知识。该问题的传统方法使用一种称为“ Feynman图表”的技术。为了解决散射幅度问题，人们总结了（许多）相关的Feynman图。大约十年前，普林斯顿州高级研究所的Nima Arkani Hamed周围的一群物理学家启动了一项计划，以更加整体的方式重新制定了散射幅度问题的解决方案。令人惊讶的是，他们的方法将主题与纯数学的许多主题联系在一起，其中候选人是专家，尤其是群集代数和有限维代数的代表理论。候选人的提议是两个方面的提议。物理学家对散射幅度问题问题的方法围绕着几何对象的构建，该对象包含其中编码的答案的基本要素。对于正在讨论的量子场理论，这需要建立一系列复杂性的多面体。该系列中的第0-多面体是Associahedron，最初是由James Stasheff在1960年代描述的，但由于其与群集代数的联系，最近看到了新的兴趣。候选人与合作者一起表明，该系列中的下一个多层也连接到集群代数。候选人建议使用群集代数（及相关有限维代数）来构建所有所需的多面体，还可以使用类似的非多层空间。该程序将为群集代数和有限二重要代数的代表理论提供新的启示，以及在散射散射增长方面的前进研究。