A review of the concept of Complexity as have emerged in studying long time properties of systems including blackholes will be presented. Some emphasis will be made on describing features of a particular form of Complexity, the Krylov Complexity. These include its features when interpolating among various types of systems: free, strongly integrable and chaotic ones. Finally I will present a case where we have shown that the value of the Krylov Complexity, a quantum information concept, is equal to the value of a geodesic in a certain background, a geometric concept.

Recent catalog of Faraday rotation measures (RM) of extragalactic sources together with the synchrotron polarization data from WMAP and Planck provide us with the wealth of information on the Galactic magnetic field (GMF). In this talk, we will present a new model of the regular GMF outside of the thin disk. The model is based on several phenomenological components of the GMF -- the spiral arms, the toroidal halo, the X-shaped field and the compressed field of the Local Bubble wall.

In General Relativity (GR), including Einstein-Maxwell theory, it is remarkable that all asymptotically flat black hole (BH) solutions have vanishing Love numbers. Consequently, the Love numbers of BHs present an excellent opportunity to examine any deviations from GR. We will investigate tidal deformations concerning neutral BHs and extremal BHs in EFT of gravity. In four dimensions, the primary contribution to the tidal Love numbers of neutral BHs arises from six derivative operators, while the Love numbers of extremal BHs are subject to corrections from four derivative operators.

We will discuss two aspects of black hole perturbation theory: symmetries of static perturbations around black holes which underpins the vanishing of the tidal Love numbers, and nonlinear quasi-normal modes during black hole ringdown. 

Dark matter in the Universe can be considered as a collisionless self-gravitating fluid obeying the Vlasov-Poisson equations. In the standard picture of cosmic structure formation, the first dark matter objects to form are expected to be microhalos of roughly Earth mass and solar system size. These halos can subsequently merge to form larger dark matter halos such as that of our Galaxy. In practice, resolving dark matter dynamics relies on a N-body approach, but with the advent of exaflopic computers it now becomes possible to solve directly Vlasov dynamics in six-dimensional phase-space.