Dr Matias Ruiz

I am a Lecturer in Applied Mathematics at the University of Leicester, a member of the Scientific Computing group and the PGR Tutor for Mathematics. 

My research concerns the mathematics of wave propagation, and in particular how geometry and material properties shape wave behaviour. I combine rigorous analysis, asymptotic methods and numerical computation to study problems arising in acoustics, electromagnetism and elasticity, with applications to nanophotonics and metamaterials. Current themes include the spectral theory of boundary integral operators, scattering and resonance, and nonlocal models.

Before joining Leicester in 2023, I was a Whittaker Research Fellow at the University of Edinburgh and a postdoctoral researcher at Imperial College London. I received my PhD from École Normale Supérieure in 2017.

My research develops mathematical theory for waves in complex materials and geometries. I work at the interface of analysis, wave physics and materials science, using spectral theory, boundary integral methods and asymptotic analysis, supported by numerical computation. The applications I have in mind come from nanophotonics and metamaterials, where small structures interact with light in ways that classical models struggle to capture.

A central thread is the theory of plasmonic resonances in metallic nanostructures. These resonances are governed by the spectrum of the Neumann-Poincaré operator, which depends delicately on geometry. I study how corners, slender shapes and self-similar structures shape this spectrum, and how resonances behave as a smooth geometry degenerates into a singular one.

A second thread concerns nonlocal models. At the nanoscale, material response depends on the field over a small neighbourhood rather than at a single point. I show that short-range nonlocal laws are equivalent to higher-order PDEs with the nonlocal length as a singular-perturbation parameter, and that this structure regularises the singular behaviour classical models predict at corners and edges. I pursue this programme in nanoplasmonics, through the hydrodynamic Drude model, and in nonlocal elasticity.

I also work on the spectral theory of scattering problems, in particular the completeness of resonant modes for Lippmann-Schwinger operators. Such results justify modal expansions widely used by physicists and engineers to design resonant devices.

I currently teach at the Leicester Dalian Instiute (DLI):

• MA2252 Introduction to Computing (first semester)
• MA2261 Linear Statistical Models (second semester)