Random matrices

…billiards, many-body systems and random matrices distinguish chaotic and integrable behaviour. Applications address geometric conjectures for complexity in quantum gravity and possible failure of effective field theory near spacelike singularities.

Joel Tropp studies structured dimension reduction through the injectivity of random maps, motivated by fast low-rank approximation and least-squares regression. This viewpoint sharpens guarantees for sparse maps and gives exponential improvements for tensor-product dimension reduction. Experiments assess the resulting structured random matrices on synthetic problems and scientific applications. Joint work with Chris Camaño, Ethan Epperly, and Raphael Meyer, available as arXiv:2508.21189.

A common approach to model local neural circuits is to assume random connectivity. But how is our choice of randomness informed by known network properties? And how does it affect the network's behavior? Previous approaches have focused on prescribing increasingly sophisticated statistics of synaptic strengths and motifs. However, experimental data on parallel dynamics of neurons is more readily accessible than their microcircuitry. We therefore propose a paradigm shift, specifying connectivity in the space that directly controls the dynamics – the space of eigenmodes. We develop a theory for