Numerical Methods in Noncommutative Matrix Analysis

The project will develop algorithms that are useful for modern physics and quantum information. In particular, the methods developed will be important for modeling topological lasers which are applicable to photonic chips and quantum circuitry. These applications pose challenges to the computational science and in particular to the linear algebra algorithms so they can work with joint measurement in multivariable setting and incommensurate observables recognizing the theoretical limits that exist.

The project will consider a variety of multivariable linear algebra algorithms, mainly those filling an immediate need in computational quantum physics and quantum information, but also those that can increase the speed and accuracy of computer shape analysis. The project will involve students and provide training in interdisciplinary projects.

This project will develop numerical methods for collections of matrices and operators arising in quantum physics and image analysis. This includes algorithms that work with finite-dimensional approximations to infinite-dimensional systems, leading to better computer modeling of quasicrystals, amorphous systems and periodic systems with defects. Methods and algorithms to be developed will be useful in the study of topological insulators, including periodically-driven systems.

The project will study various forms of spectra, including variations of the local density of states, a standard tool in many areas of physics and chemistry. The anticipated work on K-theory is expected to produce new and better tools that can be used by theoretical physicists in numerical studies. At the core of these methods is the study of joint approximate eigenvectors.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.