Section 4 Seminar, Autumn 2025

In periodic media, the emergence of band gaps leads to frequency ranges where waves are evanescent, providing the fundamental mechanism behind localisation, guided modes, and topologically protected transport. In this talk, we extend the traditional concept of band structures to the complex band structure for the Helmholtz scattering problem, showing that it provides a quantitative description of decay behaviour in defective and non-Hermitian periodic systems. Furthermore, we present a universal spectral theory for banded Toeplitz and tridiagonal k-Toeplitz operators, offering a unifying framework with broad applications across physics and engineering.

This talk is about recent works on short wave-long wave interactions in the relativistic context. In particular, we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless $1+3$-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the $1+3$-dimensional relativistic Euler equations. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this work is a clear formulation of the relativistic Lagrangian transformation. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This is a joint work with Joao Paulo Dias.