Combinatorics and geometry of mutations

这个项目的目标是更好地理解一种叫做“突变”的矩阵操作。矩阵是排列在正方形网格中的数字的集合。矩阵突变是将一个矩阵按照一定的规则进行改变，从而得到一个新的矩阵。这些规则本身似乎很奇怪，而且违反直觉，但在许多非常重要的数学领域，包括teichmller理论、泊松几何、颤振表示、李论、代数几何、代数组合学，甚至偏微分方程（在描述浅水波的方程中），矩阵突变正在幕后发生。矩阵突变出现在某些类型的代数（在称为“聚类代数”的设置中操作多变量函数），组合学（通过离散对象建模聚类代数）和几何（测量二维空间中的距离）中。该项目的一个关键组成部分是研究矩阵突变，重点是“突变扇”。扇子是一种将空间切割成碎片的方法（有一定的规则）。例如，如果我们在xy平面上通过（0,0）画三条不同的线，它们将空间切割成六块，这些块定义了一个扇形。如果我们把两个紧挨着的片段粘在一起做成一个片段，得到的五个片段就是一个简单的突变扇的例子。项目的一部分涉及通用几何系数和变异扇。具体来说，目标是构建由仿射型Cartan矩阵和曲面产生的矩阵的突变扇，并在这些情况下构建通用系数。此外，该研究将确定突变迷和其他迷之间的关系，如热带化簇代数、半不变量迷和散射图。项目的第二部分涉及交换矩阵上的优势关系。如果矩阵B具有相同的（弱）符号模式并且B的每一项都弱大于B‘的相应项，则矩阵B’优于矩阵B'。这部分项目的目标是理解在许多例子中观察到的现象，即(1)矩阵之间的优势关系经常导致突变扇之间的细化关系，(2)两个聚类代数之间经常存在代数关系。该项目的第三部分涉及仿射型的聚类代数和Coxeter-Catalan组合。本文的目标是构建簇代数的仿射型几乎正根模型的类似物，并将它们与仿射双寒武纪扇联系起来。

The goal of this project is to better understand an operation on matrices called "mutation." A matrix is a collection of numbers arranged in a square grid. Matrix mutation takes a matrix and changes it according to certain rules to make a new matrix. The rules themselves seem strange and counterintuitive, but matrix mutation is happening behind the scenes in many very important mathematical areas, including Teichmüller theory, Poisson geometry, quiver representations, Lie theory, algebraic geometry, algebraic combinatorics, and even partial differential equations (in the equations describing shallow water waves). Matrix mutation appears in certain kinds of algebra (manipulating multivariable functions in a setting called a "cluster algebra"), combinatorics (modeling cluster algebras by discrete objects) and geometry (measuring distances in two-dimensional spaces). A key component of this project is to study matrix mutation, with a focus on the "mutation fan." A fan is a way of cutting space into pieces (subject to certain rules). For example, if we draw three different lines through (0,0) in the xy-plane, they cut space into six pieces, and those pieces define a fan. If we take two of the pieces that are right next to each other and glue them together to make one piece, the resulting five pieces are a simple example of a mutation fan. One part of the project concerns universal geometric coefficients and mutation fans. Specifically, the goal is to construct the mutation fan for matrices arising from Cartan matrices of affine type and from surfaces, and to construct universal coefficients in those cases. Furthermore, the research will pin down the relationships between mutation fans and other fans, like tropicalized cluster algebras, fans of semi-invariants, and scattering diagrams. A second part of the project concerns the dominance relation on exchange matrices. A matrix B dominates a matrix B' if they have the same (weak) sign pattern and each entry of B is weakly larger than the corresponding entry of B'. The goal of this part of the project is to understand a phenomenon observed in many examples, namely (1) that dominance relations among matrices often lead to refinement relations among mutation fans and (2) that there is often an algebraic relationship between the two cluster algebras. The third part of the project concerns cluster algebras and Coxeter-Catalan combinatorics in affine type. Here the goal is to construct the affine-type analogs of almost-positive root models for cluster algebras, and to relate them to affine doubled Cambrian fans.