The study of constant scalar curvature Kähler (cscK) metrics on compact complex manifolds is a classical topic that has attracted enormous interest since the 1950s. However, detecting the existence of cscK metrics is a difficult task, which in the projective integral case conjecturally amounts to proving an important algebro-geometric stability notion (K-stability). Recent significant advancements have established that the existence of unique cscK metric in a Kähler class is equivalent to the coercivity of the so-called Mabuchi functional. I will extend the notion of cscK metrics to singular varieties, and I will show the existence of these special metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. A key point in this variational approach is the lower semicontinuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact Kähler varieties with klt singularities. The latter strengthens evidence supporting the openness of (uniform) K-stability for general families of normal compact Kähler varieties with klt singularities. This is a joint work with Chung-Ming Pan and Tat Dat Tô.

Interpolation of differential forms is a challenging aspect of modern approximation theory. Not only does it shed new light on some classical concepts of interpolation theory, such as the Lagrange interpolation and the Lebesgue constant, but it also suggests that they can be extended to a very general framework. As an extent of that, it is worth pointing out that the majority of classical shape functions commonly used in finite element methods, such as those involved in Nedelec or Raviart-Thomas elements, can be seen as a specialisation of this theory. Of course, this generality brings along the evident downside of an unfriendly level of abstraction. The scope of this series of two seminars is thus twofold: presenting the main challenges of this branch of approximation theory but in a concrete manner. The first seminar will hinge on a development of a convenient one dimensional toy model that enlightens parallelisms and differences with usual nodal interpolation. In the second seminar will extend these techniques to the multi-dimensional framework, motivating our choices by a geometrical flavour. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).