Chemical Evolution in the Milky Way: Rotation-based ages for
  APOGEE-Kepler cool dwarf stars
  We use models of stellar angular momentum evolution to determine ages for
$\sim500$ stars in the APOGEE-\textit{Kepler} Cool Dwarfs sample. We focus on
lower main-sequence stars, where other age-dating tools become ineffective. Our
age distributions are compared to those derived from asteroseismic and giant
samples and solar analogs. We are able to recover gyrochronological ages for
old, lower-main-sequence stars, a remarkable improvement over prior work in
hotter stars. Under our model assumptions, our ages have a median relative
uncertainty of $14\%$, comparable to the age precision inferred for more
massive stars using traditional methods. We investigate trends of galactic
$\alpha$-enhancement with age, finding evidence of a detection threshold
between the age of the oldest $\alpha$-poor stars and that of the bulk
$\alpha$-rich population. We argue that gyrochronology is an effective tool
reaching ages of 10--12 Gyr in K- and early M-dwarfs. Finally, we present the
first effort to quantify the impact of detailed abundance patterns on
rotational evolution. We estimate a $\sim15\%$ bias in age for cool,
$\alpha$-enhanced (+ 0.4 dex) stars when standard solar-abundance-pattern
rotational models are used for age inference, rather than models that
appropriately account for $\alpha$-enrichment.
