Understanding limits on the superconducting transition temperature Tc is a problem of fundamental importance. I will discuss rigorous upper bounds on the superfluid phase stiffness for interacting,

multi-band systems, valid in any dimension. This in turn leads to upper bounds [1] on Tc in 2D that hold irrespective of pairing mechanism, interaction strength, or order parameter symmetry, so long as one has a BKT transition. Our bounds are particularly useful for the strongly correlated regime of low density, narrow band systems, where mean-field theory fails. For a simple parabolic band in 2D with Fermi energy Ef, we find that Tc can never exceed Ef/8, an exact result that has direct implications for the 2D BCS-BEC crossover in ultracold Fermi gases. I will then discuss the application of our multi-band bounds to monolayer FeSe/STO and to Magic-Angle Twisted Bilayer Graphene, where we find very interesting results for Tc constrained by the band structure. Finally, I will discuss the question of deriving rigorous upper bounds on Tc in 3D.

[1] T. Hazra, N. Verma, M. Randeria, PRX 9, 031049 (2019)