Friday, 16 February 2024, 11:00–17:00, Room 5159.0062 in the Energy Academy Europe building (Groningen)
Please register your participation here. The registration is free and will help us to arrange the catering.
On Neumann-Poincaré operators and self-adjoint transmission problems [slides]
In this talk, we discuss the self-adjointness in $L^2$-setting of the operators acting as $−\mathrm{div} \cdot h\nabla$, with piecewise constant functions $h$ having a jump along a Lipschitz hypersurface $\Sigma$, without explicit assumptions on the sign of $h$. We establish a number of sufficient conditions for the selfadjointness of the operator with $H^{\frac{3}{2}}$-regularity in terms of the jump value and the regularity and geometric properties of $\Sigma$. An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on $\Sigma$, which is new for the Lipschitz setting. Based on joint work with Konstantin Pankrashkin.
Cheeger's inequality in Carnot-Carathéodory spaces [slides]
We explore extending the classical Cheeger inequality to Carnot-Carathéodory (CC) spaces, which are manifolds in which shortest paths can only take velocities confined to a sub-bundle of the tangent bundle. During the talk, I will discuss CC-spaces and their basic properties, as well as the key geometric and analytic concepts used in the proof of Cheeger's inequality. Based on [arXiv:2312.13058].
The wave resolvent for compactly supported perturbations of static spacetimes [slides]
I will give an elementary microlocal proof of the essential self-adjointness of the Lorentzian Laplace-Beltrami operator $P$ in the case of compactly supported perturbations of static spacetimes. A modification of the proof also yields uniform microlocal estimates for the resolvent, which serve to prove a relationship between functions of $P$ (in particular the complex powers) and local geometric invariants (following a similar result due to Dang-Wrochna).
Organizers: Konstantin Pankrashkin (Oldenburg), Marcello Seri (Groningen) and Michal Wrochna (Utrecht)