Kepler's Discoveries Will Continue: 21 Important Scientific Opportunities with Kepler & K2 Archive Data

NASA's Kepler Space Telescope has collected high-precision, high-cadence time series photometry on 781,590 unique postage-stamp targets across 21 different fields of view. These observations have already yielded 2,496 scientific publications by authors from 63 countries. The full data set is now public and available from NASA's data archives, enabling continued investigations and discoveries of exoplanets, oscillating stars, eclipsing binaries, stellar variability, star clusters, supernovae, galaxies, asteroids, and much more. In this white paper, we discuss 21 important data analysis projects which are enabled by the archive data. The aim of this paper is to help new users understand where there may be important scientific gains left to be made in analyzing Kepler data, and to encourage the continued use of the archives. With the TESS mission about to start releasing data, the studies will inform new experiments, new surveys, and new analysis techniques. The Kepler mission has provided an unprecedented data set with a precision and duration that will not be rivaled for decades. The studies discussed in this paper show that many of Kepler's contributions still lie ahead of us, owing to the emergence of complementary new data sets like Gaia, novel data analysis methods, and advances in computing power. Kepler's unique data archive will provide new discoveries for years to come, touching upon key aspects of each of NASA's three big astrophysics questions; How does the universe work? How did we get here? Are we alone?