Combinatorial aspects of representation theory and geometry

组合学是研究离散排列的学科，比如将n个球分成k个不同盒子的不同方法。通常，我们有可能理解这种离散的安排，而不是我们希望理解我们在现实生活中通常遇到的不断变化的现象。然而，事实证明，离散结构在现实生活和其他数学领域都有重要的意义。我的建议包括使用组合技术来理解代数和物理中出现的数学结构。我感兴趣的物理学是计算散射振幅的问题。这是一个非常自然和基本的问题。我们把一堆粒子扔向对方。它们以某种方式相互作用，然后缩小。我们想知道结果。因为我们是在量子环境中，结果是不确定的，但有不同的可能性，每种可能性都有一个概率。通常计算散射振幅的方法是写下典型的大量费曼图，这些图编码了粒子之间可能的相互作用，然后将每个粒子的贡献加起来。令人惊讶的是，当我们这样做的时候，答案是简单而对称的，而单独的求和项却不是这样。这表明，费曼图方法可能掩盖了正在发生的事情的基本简单性。Nima Arkani-Hamed和他的合作者开发了一种测量散射振幅的新方法。在他们的方法中，费曼图被物理学家称为“壳上图”的东西所取代，但这已经被Alex Postnikov和其他数学家以“塑性图”的名义研究过了。通过与Nima Arkani-Hamed和Jaroslav Trnka合作，研究与这些平面图相关的组合学和几何，我希望能更好地理解这种新方法。组合学也可以应用于代数设置。一种方法是从一个组合系统开始并将一些代数与它联系起来。我们对组合学的理解可能会帮助我们理解这些代数。强大的代数工具也有可能提高我们对组合学的理解。更具体地说，我对反射群的组合学（置换的一种概括）和预投影代数及其商的表示理论之间的联系感兴趣。**

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. **