Combinatorial aspects of representation theory and geometry

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Combinatorics is the study of discrete arrangements, such as the different ways of dividing n balls among k different boxes. Often, it is possible to understand such discrete arrangements much better than we could hope to understand the continuously changing phenomena that we typically encounter in real life. However, as it turns out, discrete structures can have important implications, in real life, and also in other areas of mathematics. My proposal consists of using combinatorial techniques to understand mathematical structures which arise in algebra and physics. The physics I am interested in is the problem of calculating scattering amplitudes. This is quite a natural and fundamental problem. We throw a collection of particles at each other. They interact somehow, and then zoom off. We want to know the outcome. Because we are in a quantum setting, the outcome is uncertain, but there are different possibilities, to each of which a probability can be assigned. The usual approach to calculating scattering amplitudes is to write down what is typically a very large number of Feynman diagrams which encode the possible interactions among the particles, and then add up a contribution from each. It is a surprising fact that when we do this, the answer is simple and symmetrical in a way that the individual terms being summed are not. This suggests that the Feynman diagram approach might be obscuring the fundamental simplicity of what is happening. A new approach to scattering amplitudes has been developed by Nima Arkani-Hamed and a team of collaborators. In their approach, Feynman diagrams are replaced by something which the physicists call "on-shell diagrams", but which had already been studied by Alex Postnikov and other mathematicians under the name of "plabic graphs". By studying the combinatorics and geometry associated to these plabic graphs, in collaboration with Nima Arkani-Hamed and Jaroslav Trnka, I hope to come to a better understanding of this new approach. Combinatorics can also be applied in algebraic settings. One way to do this is to start from a combinatorial system and associate some kind of algebra to it. Our understanding of the combinatorics will likely help us to understand such algebras. It is also possible that powerful algebraic tools may then also improve our understanding of the combinatorics from which we set out. To be a bit more specific, I am interested in links between the combinatorics of reflection groups (a generalization of permutations) and the representation theory of preprojective algebras and their quotients.