The Panchromatic Hubble Andromeda Treasury XI: The Spatially-Resolved
  Recent Star Formation History of M31
  We measure the recent star formation history (SFH) across M31 using optical
images taken with the \texit{Hubble Space Telescope} as part of the
Panchromatic Hubble Andromeda Treasury (PHAT). We fit the color-magnitude
diagrams in ~9000 regions that are ~100 pc $\times$ 100 pc in projected size,
covering a 0.5 square degree area (~380 kpc$^2$, deprojected) in the NE
quadrant of M31. We show that the SFHs vary significantly on these small
spatial scales but that there are also coherent galaxy-wide fluctuations in the
SFH back to ~500 Myr, most notably in M31's 10-kpc star-forming ring. We find
that the 10-kpc ring is at least 400 Myr old, showing ongoing star formation
over the past ~500 Myr. This indicates the presence of molecular gas in the
ring over at least 2 dynamical times at this radius. We also find that the
ring's position is constant throughout this time, and is stationary at the
level of 1 km/s, although there is evidence for broadening of the ring due to
diffusion of stars into the disk. Based on existing models of M31's ring
features, the lack of evolution in the ring's position makes a purely
collisional ring origin highly unlikely. We find that the global SFR has been
fairly constant over the last ~500 Myr, though it does show a small increase at
50 Myr that is 1.3 times the average SFR over the past 100 Myr. During the last
~500 Myr, ~60% of all SF occurs in the 10-kpc ring. Finally, we find that in
the past 100 Myr, the average SFR over the PHAT survey area is $0.28\pm0.03$
M$_\odot$ yr$^{-1}$ with an average deprojected intensity of $7.3 \times
10^{-4}$ M$_\odot$ yr$^{-1}$ kpc$^{-2}$, which yields a total SFR of ~0.7
M$_\odot$ yr$^{-1}$ when extrapolated to the entire area of M31's disk. This
SFR is consistent with measurements from broadband estimates. [abridged]
