Emergent Topology and K-Theory of Matrix Models

我们引导光波、电子或声波的能力正在迅速增长，这是由于受物质系统拓扑属性控制的运动模式受到保护。 通过拓扑保护引导光由于其经济和科学重要性而特别令人兴奋。 拓扑保护模式的理论描述通常使用非标准形式的拓扑结构，用矩阵模型来描述。 该项目旨在发展矩阵模型的理论，以及算子代数的更广泛的理论主题，其灵感来自物理学的进步。 它还打算使知识从该地区的运营商代数提供给物理学家的形式的算法和公式适用于光波和电子波。 该项目旨在在数学分析工作与当前基于石墨烯的电子学和基于硅的光子学研究之间建立异常直接的联系。 然而，重点将是一个理论的矩阵模型的对称性，是强大的，足以同样很好地与各种类型的系统，并在矩阵理论的进展，将支持未来的应用和未来的理论工作在该地区的分析称为非交换拓扑结构。 允许在流形内跳跃的自由费米子系统的模型可以自然地形成为一个群，该群抽象同构于与流形相关联的更标准的群。 这意味着算子代数中常用的同调理论有一个新的表述。 这个项目的核心将是找到这样一个可计算的重构同调理论的整个类别的算子代数有零，一个或两个增加的对称性。 现在需要数值算法来快速分析许多这些矩阵模型。 虽然几何学中的经典矩阵不变量可以通过手工计算来处理小问题，但物理学家现在需要在大量的大矩阵上计算这些不变量。 通过改进对这些不变量的数学理解，可以提高这种计算的进展。 例如，在驱动光子系统中，研究人员将Bott不变量用于几乎可交换酉矩阵，这是有很好的理论基础的。 然而，有一个相关的不变量，基于矩阵模型的涌现拓扑，可以通过更快的算法计算。 这个不变量将被研究，目的是把它放在一个更好的理论背景下。 凝聚态和其他物理系统的涌现拓扑也将用于增强矩阵模型结构的可视化。

Our ability to guide waves of light, electrons, or sound is growing rapidly, due to protected modes of motion governed by topological attributes of material systems. Guiding light via topological protection is particularly exciting due to its economic and scientific importance. Theoretical descriptions of topologically protected modes often use non-standard forms of topology, described in terms of matrix models. The project aims to develop the theory of matrix models, and the broader theoretical subject of operator algebras, in ways that are inspired by the advances in physics. It also intends to make knowledge from the area of operator algebras available to physicists in the form of algorithms and formulas applicable to light waves and electronic waves. The project aims to develop an unusually direct connection between this work in mathematical analysis and current research on graphene-based electronics and silicon-based photonics. However, the emphasis will be on a theory of matrix models with symmetries that is robust enough to work equally well with various types of systems, and on advances in matrix theory that will support future applications and future theoretical work in the area of analysis known as noncommutative topology. Models of free fermionic systems that allow hopping within a manifold can be naturally formed into a group that is abstractly isomorphic to more standard groups associated to a manifold. This suggests that there is a reformulation of the common homology theories used in operator algebras. Central to this project will be finding such a computable reformulation of homology theories for the whole categories of operator algebras that have zero, one, or two added symmetries. There is now a need for numerical algorithms to analyze many of these matrix models quickly. Although the classical matrix invariants from geometry can be dealt with by hand calculations for small problems, physicists now require calculation of these invariants on multitudes of large matrices. Progress in such calculations can be enhanced by improved mathematical understanding of these invariants. For example, researchers in driven photonic systems use the Bott invariant for almost commuting unitary matrices, which is on good theoretical grounds. However, there is a related invariant, based on the emergent topology of a matrix model, that can be computed by a much faster algorithm. This invariant will be studied, with a goal of putting it in a better theoretical context. The emergent topology of condensed matter and other physical systems will also be used to enhance visualization of the structure of matrix models.