Algebra, combinatorics and mathematical computer science

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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.