Project Grant 2550762

The National Science Foundation Division of Mathematical Sciences awarded the University of California, Los Angeles $145,000 on August 1, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to develop rigorous mathematical theory predicting localization and delocalization transitions in complex disordered quantum systems.

The research focuses on non-mean-field random matrices and addresses fundamental questions about when electrons, waves, and other excitations remain trapped in a small region versus spreading throughout an entire material. The project builds on recent advances in the delocalization conjecture for random band matrices and will connect several influential models of disordered quantum systems within a common framework, identifying the parameters that govern transport behavior. The investigator will establish polynomial-scale delocalization bounds through resolvent analysis, local eigenvalue statistics, and eigenvector localization lengths, with particular emphasis on proving the delocalized regime of a universal coupling-strength conjecture for general block random Schrödinger operators including random band matrices, block Anderson and Wegner orbital models, and operators with long-range variance profiles.

The project will translate its graphical methods into a new course, student research projects, and open-source computational tools, expanding access to modern ideas at the intersection of probability, physics, and computation.

Performance occurs in Los Angeles, California. The period of performance runs from August 1, 2026, through July 31, 2028. This is a project grant, a form of assistance funding.