  As the closest example of a galactic nucleus, the Galactic center (GC)
presents an exquisite laboratory for learning about supermassive black holes
(SMBH) and their environment. We describe several exciting new research
directions that, over the next 10 years, hold the potential to answer some of
the biggest scientific questions raised in recent decades: Is General
Relativity (GR) the correct description for supermassive black holes? What is
the nature of star formation in extreme environments? How do stars and compact
objects dynamically interact with the supermassive black hole? What physical
processes drive gas accretion in low-luminosity black holes? We describe how
the high sensitivity, angular resolution, and astrometric precision offered by
the next generation of large ground-based telescopes with adaptive optics will
help us answer these questions. First, it will be possible to obtain precision
measurements of stellar orbits in the Galaxy's central potential, providing
both tests of GR in the unexplored regime near a SMBH and measurements of the
extended dark matter distribution that is predicted to exist at the GC. Second,
we will probe stellar populations at the GC to significantly lower masses than
are possible today, down to brown dwarfs. Their structure and dynamics will
provide an unprecedented view of the stellar cusp around the SMBH and will
distinguish between models of star formation in this extreme environment. This
increase in depth will also allow us to measure the currently unknown
population of compact remnants at the GC by observing their effects on luminous
sources. Third, uncertainties on the mass of and distance to the SMBH can be
improved by a factor of $\sim$10. Finally, we can also study the near-infrared
accretion onto the black hole at unprecedented sensitivity and time resolution,
which can reveal the underlying physics of black hole accretion.
