The explicit computation of the spectrum of the Laplacian on closed Riemannian manifolds is a challenging task that only succeeds under strong symmetry assumptions. After the classical examples of spheres, projective spaces, and flat tori, Riemannian symmetric spaces G/K were the first large class of manifolds for which the spectrum could be computed explicitly via representation theoretic tools.

For homogeneous non-symmetric spaces, the classical approaches fail. I will give an introduction to new methods for computing spectra on large families of naturally reductive homogeneous metrics. Many examples like Aloff-Wallach manifolds or Berger spheres will be used to illustrate the results; in fact, explicit spectra were computed using Python and are available as a Jupyter notebook.

This is joint work with Jonas Henkel and Leandro Cagliero.

The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting two parallel lines in the euclidean plane perpendicularly. Straight lines connecting these orthogonally are obviously equilibria and absolute minimisers of the elastic energy.
It is proved that these have, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level $1.9615\, \pi$. It was originally conjectured that this level should be $2\pi$, which is the energy of a half period of Euler's elastica, which is unstable and (up to scaling, translation and rotation) the only nontrivial equilibrium.
The method of proof cannot be pushed to the previously conjectured level, and numerical simulations give evidence that this conjecture may in fact be false.
The core of the argument is an interpolation inequality which looks purely analytical. However, the necessary conditions on possibly optimal functions turn out to be the elastica equations with respect to the euclidean, spherical or hyperbolic metric, respectively.
This talk reports on joint work with Klaus Deckelnick, Robert Nürnberg (numerical analysis & experiments), and Glen and Valentina-Mira Wheeler (geometric analysis).

This talk concerns the Willmore flow for graphs over a bounded planar domain with fixed Dirichlet boundary values. I will present a low-regularity theory that avoids the classical fourth-order compatibility condition at the initial time. The main analytic ingredient is the use of weighted parabolic Hölder spaces, which allow one to derive the a priori estimates required for a linearization-based proof of short-time existence. I will then discuss global gradient bounds and exponential convergence for sufficiently small initial data. This problem provides a model setting for higher-order geometric flows with fixed boundary data.

Mean curvature flow is classically understood as the $L^2$-gradient flow of the volume functional. In the non-compact setting, however, this variational interpretation degenerates, since the volume is typically infinite and the associated energy identity becomes vacuous. In this talk, we present a new variational framework for non-compact mean curvature flow by interpreting it as the gradient flow of a calibration energy that quantifies the deviation from calibrated geometry. We also discuss geometric applications.