Hot Mathematics in Cold Physics

We briefly discuss the common physical features of peculiar phenomena of quantum ultra-cold matter, as, e.g., Bose-Einstein condensation, superfluidity and superconductivity, as well as the mathematics of the corresponding models used in physics. We stress the role of effective theories and comment on their mathematical derivation. We then focus on the specific example of the Ginzburg-Landau theory of superconductivity and the phenomenon of the response of a superconducting material to an external magnetic field. We review some recent results on the surface superconductivity regime and in particular describe how the boundary curvature affects the energy and distribution of Cooper pairs in the material.