Algebra, combinatorics and mathematical computer science

物理学中的散射振幅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重要性，而且具有实际意义：为了分析来自粒子加速器（如大型强子对撞机）的数据，需要精确的散射振幅知识。解决这个问题的传统方法使用一种叫做“费曼图”的技术。为了解决散射振幅问题，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的尼玛·阿卡尼-哈米德（Nima Arkani-Hamed）周围的一群物理学家发起了一个项目，以一种更全面的方式重新定义散射振幅问题的解决方案。令人惊讶的是，他们的方法将该主题与候选人是纯数学专家的许多主题联系起来，特别是聚类代数和有限维代数的表示理论。候选人的建议是双重的：他将解决来自物理学家方法的新的和令人兴奋的数学问题，他将积极地与物理学家合作，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一个几何物体的构造，这个几何物体包含了编码在其中的答案的基本元素。对于正在讨论的量子场论，这需要构造一系列越来越复杂的多面体。这个系列中的第0个多面体是结合面体，最初是由James Stasheff在20世纪60年代描述的，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，候选人已经证明了该系列中的下一个多面体也与聚类代数相连。候选人建议使用聚类代数（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。该计划将为簇代数和有限维代数的表示理论提供新的亮点，并推动散射振幅的研究。

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.