realized in microscopic calculations based on dynamical mean-field theory [69, 70], which also start with the assumption that the self-energy is momentum independent. At a microscopic level, it is understood that the renormalizations in the heavy-fermion lattice are closely connected with the Kondo effect of the magnetic ions in the bath of conduction electrons. This physics is well-incorporated in the models of the lattice by a variety of different techniques, the most versatile of which is the dynamical mean-field theory which yields results similar to Eqs. (36) together with the frequency dependence of the one-particle spectra. A theory of two interacting Kondo impurities [71] with interactions among the magnetic ions competing with the Kondo effect at either also exists. The theory yields a low energy effective Hamiltonian for the problem with several interesting terms, including pairing interactions among the quasi-particles. A low energy Hamiltonian for the lattice at a similar level has not yet been derived but is essential for a detailed theory of superconductivity in them. As discussed above, one can motivate the d-wave symmetry of superconductivity in most of the heavy-fermions through exchange of antiferromagnetic fluctuations. But this does not answer the question of the energy scales of fluctuations in the critical regime. All we know is just that the renormalizations discussed through Eqs.(36) give the scale of the upper limit of the cut-off energy of the pairing fluctuations to be down by z from the bare parameters, i.e. of O ( E$_{f}$ ), the renormalized fermi-energy. (As discussed already, the value of z serves to renormalize the cutoff scale as well as the coupling constants in general accord with Landau fermi-liquid theory.) The actual experimental values of T$_{c}$ are down from this by only between one and two orders of magnitude from the upper cut-off, see Fig. (10). One may speculate that this is associated with the fact that unlike liquid He $^{3}$, superconductivity in heavy-fermions generally always occurs in the region around quantum- criticality, see Fig.(9), and the criticality is not of the conventional variety where the energy scale of fluctuations → 0 as criticality is approached. In the one case in which quantum-critical fluctuations have been studied in single-crystals by neutron scattering the criticality is of the topological or local variety [72] discussed in Sec. II: the fluctuation spectra scales as ω/T but is essentially q-independent; the space and time are not tied together as in Gaussian quantum-criticality. This particular compound is however not superconducting. But, based on the measured specific heat, and resistivity measurements, one can be fairly sure that criticality is of the same variety in the heavy fermions where superconductivity occurs with relatively large values of T$_{c}$/E$_{f}$ . The measured 41