Small Banach spaces C(K) without nice renormings

When dealing with vector spaces C(K) of continuous real-valued functions on compact spaces K, one by default endows them with the supremum norm, making those spaces Banach. This norm is particularly handy and simple in use, however it has bad geometric properties such as, e.g., the existence of non-degenerate intervals in the spheres. Fortunately, in many cases the supremum norm can be exchanged for a better norm, with much nicer, more rotund or strictly convex features, yet still preserving the topological structure of a given space C(K), which is crucial for applications in such areas as variational calculus or optimization theory. For example, this happens when C(K) is separable or K is scattered of finite Cantor-Bendixson height. On the other hand, it is known that the spaces C(omega*) and C(beta omega_1) fail such renormings. The latter spaces have, however, large densities (>=2^omega) and so the natural question is whether there are spaces C(K) without good renormings and of small uncountable density (e.g. omega_1<2^omega). During my talk I'll show that consistently this can be the case, presenting various examples obtained e.g. by means of Suslin trees or using ground model sigma-complete Boolean algebras in side-by-side Sacks extensions.