Spectral estimates on manifolds have been studied since the end of the 80s, in connection with spectral geometry and control of the heat equation. After introducing some background and motivation, I will present an original spectral estimate result on the hyperbolic half-plane, the first of its kind in a non-Euclidean setting.
If time allows, the proof will be discussed along with possible generalizations. The tools involved come from spectral theory, harmonic analysis, control of PDEs and basic geometry.