  The Galactic Center (GC) is dominated by the gravity of a super-massive black
hole (SMBH), Sagittarius A$^*$, and is suspected to contain a sizable
population of binary stars. Such binaries form hierarchical triples with the
SMBH, undergoing Eccentric Kozai-Lidov (EKL) evolution, which can lead to high
eccentricity excitations for the binary companions' mutual orbit. This effect
can lead to stellar collisions or Roche-lobe crossings, as well as orbital
shrinking due to tidal dissipation. In this work we investigate the dynamical
and stellar evolution of such binary systems, especially with regards to the
binaries' post-main-sequence evolution. We find that the majority of binaries
(~75%) is eventually separated into single stars, while the remaining binaries
(~25%) undergo phases of common-envelope evolution and/or stellar mergers.
These objects can produce a number of different exotic outcomes, including
rejuvenated stars, G2-like infrared-excess objects, stripped giant stars, Type
Ia supernovae (SNe), cataclysmic variables (CVs), symbiotic binaries (SBs), or
compact object binaries. We estimate that, within a sphere of 250 Mpc radius,
about 7.5 to 15 Type Ia SNe per year should occur in galactic nuclei due to
this mechanism, potentially detectable by ZTF and ASAS-SN. Likewise we estimate
that, within a sphere of 1 Gpc$^3$ volume, about 10 to 20 compact object
binaries form per year that could become gravitational wave sources. Based on
results of EKL-driven compact object binary mergers in galactic nuclei by Hoang
at al. (2018), this compact object binary formation rate translates to about 15
to 30 events per year detectable by Advanced LIGO.
