Exploring the Supermassive Black Hole at the Galactic Center and the Stars in its Environment

As the closest galactic nucleus, the Galactic center provides a unique opportunity for learning about the supermassive black hole at the center of the Milky Way, Sgr A*, and its environment. High-angular-resolution observations in the near-infrared further this advantage. This thesis presents an investigation of the long-term near-infrared accretion properties of Sgr A*, and characterizes the star formation history in this extreme environment. This work utilizes the unique data gathered by the UCLA Galactic Center Group over 25 years to answer fundamental questions about the supermassive black hole and the formation of stars around it. First, we investigate the near-infrared variability of Sgr A* with the longest time baseline yet considered. The recently improved speckle holography technique has led to the capability to detect Sgr A* in the early years of the group's monitoring program. We carry out a new analysis of speckle imaging data (1995 - 2005) obtained from the Keck observatory, enabling new detections of Sgr A* in the near-infrared over a decade that was previously inaccessible at these wavelengths. We report that the near-infrared brightness of Sgr A* and its variability are consistent over 22 years, which addresses variability timescales that are 10 times longer than earlier published studies. The power spectral density of Sgr A* shows a plateau between $\sim$80 days and 7 years, further confirming that it is uncorrelated in time beyond the previously-proposed, single power-law break of $\sim$245 minutes. We also investigate and note that the closest approach of the extended and dusty object G1, experiencing tidal disruption, had no apparent effect on the near-infrared emission from the accretion flow onto Sgr A*. Second, we report the first star-formation history study of the Milky Way's nuclear star cluster (NSC) which includes observational constraints from a large sample of stellar metallicity measurements from Gemini and VLT. Along with stellar photometry and spectroscopically derived stellar temperatures, a Bayesian inference approach is developed to derive the NSC's star formation history. By including metallicity measurements, the low-temperature red giants that were previously difficult to constraint are now accounted for, and the best fit favors a two-component model. The dominant ($\sim$93\% of the mass), metal-rich component has an age of 5$^{+3}_{-2}$ Gyr, which is likely $\sim$3 Gyr younger than earlier studies with fixed (solar) metallicity; this younger age challenges the co-evolution models of the supermassive black hole, the NSC and the bulge. The minor component ($\sim$7\%) is metal-poor and its age is uncertain though likely less than that of the dominant component. We present updated estimates of the number of compact objects at the Galactic center and their rates of mergers in order to inform interpretations of gravitational-wave detections. Of particular note, when metallicity measurements are included, the predictions result in 2 - 4 times fewer neutron stars compared to earlier predictions that assume solar metallicity, introducing a possible new path to understand the so-called ``missing pulsar problem" at the Galactic center