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.

物理学中的散射振幅问题是描述基本粒子相互散射时发生的事情的问题。这个问题不仅具有重要的理论意义，而且也具有实际意义：为了分析来自粒子加速器（如大型强子对撞机）的数据，需要精确地了解散射振幅。解决这个问题的传统方法是使用一种称为“费曼图”的技术。“为了解决散射振幅问题，人们总结了（许多）相关费曼图的贡献。大约10年前，普林斯顿高等研究院（Institute for Advanced Study in Princeton）的尼玛·阿尔卡尼-哈米德（Nima Arkani-Hamed）周围的一群物理学家发起了一个项目，以更全面的方式重新设计散射振幅问题的解决方案。他们的方法，令人惊讶的是，连接的主题到一些主题在纯数学，其中候选人是一个专家，特别是集群代数和表示理论的有限维代数。候选人的建议是双重的：他将解决新的和令人兴奋的数学问题起源于物理学家的方法，他将与物理学家积极合作，将这些结果应用于散射振幅问题。物理学家解决散射振幅问题的方法围绕着构建一个几何对象，该对象包含散射振幅的基本元素。答案就在其中。对于正在讨论的量子场论来说，这需要建造一系列越来越复杂的多面体。第0个多面体是associahedron，最初由James Stasheff在20世纪60年代描述，但最近由于它与簇代数的联系而重新引起了人们的兴趣。与合作者一起，候选人已经证明了系列中的下一个多面体也连接到一个簇代数。候选人提出使用簇代数（和相关的有限维代数）来构造所有需要的多面体，以及类似的非多面体空间。该计划将为簇代数和有限维代数的表示理论提供新的启发，并推动散射振幅的研究。