  The nuclear symmetry energy figures crucially in the structure of asymmetric
nuclei and, more importantly, in the equation of state (EoS) of compact stars.
At present it is almost totally unknown, both experimentally and theoretically,
in the density regime appropriate for the interior of neutron stars. Basing on
a strong-coupled structure of dense baryonic matter encoded in the skyrmion
crystal approach with a topology change and resorting to the notion of
generalized HLS (hidden local symmetry) in hadronic interactions, we address a
variety of hitherto unexplored issues of nuclear interactions associated with
the symmetry energy, i.e., kaon condensation and hyperons, possible topology
change in dense matter, nuclear tensor forces, conformal symmetry and chiral
symmetry etc in the EoS of dense compact-star matter. One of the surprising
results coming from the hidden local symmetry structure that is distinct from
what is given by standard phenomenological approaches is that at high density,
baryonic matter is driven by RG flow to the "dilaton-limit fixed point (DLFP)"
constrained by "mended symmetries." We further propose how to formulate kaon
condensation and hyperons in compact-star matter in a framework anchored on a
single effective Lagrangian by treating hyperons as the Callan-Klebanov
kaon-skyrmion bound states simulated on crystal lattice. This formulation
suggests that hyperons can figure in the stellar matter -- if at all -- when or
after kaons condense, in contrast to the standard phenomenological approaches
where the hyperons appear as the first strangeness degree of freedom in matter
thereby suppressing or delaying kaon condensation. In our simplified
description of the stellar structure in terms of symmetry energies which is
compatible with that of the 1.97 solar mass star, kaon condensation plays a
role of "doorway state" to strange-quark matter.
