Combinatorics and geometry of mutations

点击翻译按钮获取中文摘要

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.