Emergent Topology and K-Theory of Matrix Models

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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.