The human brain exhibits cryptic spatiotemporal dynamics that can be investigated through the lens of coupled dynamical systems and spectral decomposition methods. In this talk, I will present how concepts from harmonic analysis, Markov processes, and oscillatory dynamics provide a rigorous mathematical framework for deciphering the hidden rules governing large-scale brain organisation.

More precisely, I will show how brain patterns switch as a Markov process; how eigenmodes exhibit damped oscillatory motion; and how collective rhythms emerge from metastable synchronisation. This work illustrates a bidirectional bridge: mathematical formalisms illuminate neuroscience phenomena, while brain data offers concrete applications for advancing theoretical frameworks—ultimately enabling us to understand the dynamical principles underlying cognitive function, consciousness and mental health.

We discuss the known problem of well-posedness of singular integral operators on Morrey spaces with a special emphasis on the weighted case. In particular, we obtain a general condition on weights, which guarantees the almost everywhere convergence of singular integrals on functions in weighted generalized Morrey spaces.