Algebra, combinatorics and mathematical computer science

物理学中的散射幅度问题是描述基本粒子相互散射时会发生什么的问题。这个问题不仅具有基本的理论意义，而且也具有实际意义：为了分析来自大型强子对撞机等粒子加速器的数据，需要精确的散射幅度知识。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。大约十年前，普林斯顿高级研究所的尼玛·阿卡尼-哈米德身边的一群物理学家发起了一个项目，以更全面的方式重新定义散射幅度问题的解决方案。令人惊讶的是，他们的方法将这个话题与纯数学中的许多主题联系在一起，其中候选人是专家，特别是簇代数和有限维代数的表示理论。候选人的提议有两个方面：他将解决源自物理学家方法的新的令人兴奋的数学问题，他将积极与物理学家合作，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来说，这需要构造一系列日益复杂的多面体。该系列中的第0个多面体是缔合体，最初由James Stasheff在20世纪60年代的S中描述，但最近由于它与簇代数的联系而重新引起人们的兴趣。与合作者一起，候选人已经证明了该系列中的下一个多面体也与簇代数相连。候选人提出使用簇代数(及相关的有限维代数)来构造所需的所有多面体，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理论提供新的启示，并推动散射幅度的研究。

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.