  We briefly discuss recent experiments on the BCS-BEC crossover with ultracold
alkali-metal atoms both in three-dimensional configurations and two-dimensional
ones. Then we analyze the quantum-field-theory formalism used to describe an
attractive $D$-dimensional Fermi gas taking into account Gaussian fluctuations.
Finally, we apply this formalism to obtain a reliable equation of state of the
2D system at low temperaratures in the BEC regime of the crossover by
performing a meaningful dimensional regularization of the divergent zero-point
energy of collective bosonic excitations.
