Why is most of the star formation that goes on in the milky way occurring in the spiral arms? This fact was mentioned by our astrophysics lecturer and I can't seem to understand why. Isn't the densest and hottest part of our galaxy the central bulge? I would've thought star formation would occur primarily where there is most heat and material. 
 A: The highest density of star formation is actually in the inner region of the Milky Way. The upper panel of this figure (Fig.8 from the 2012 review article by Kennicutt & Evans) shows the estimated surface density of star formation (star formation rate per unit area, indicated by the stars) as a function of radius: you can see that this peaks in the innermost region (the "CMZ" = Central Molecular Zone, r < 250 parsecs). (Anders Sandberg's arguments about why you wouldn't have star formation in the central region of the galaxy are wrong: there's plenty of cool, dense molecular gas there.)

Nonetheless, it is reasonable to say that most of the total star formation takes place in the disk, because the disk is simply much bigger than the central region.
If you look at the star-formation-rate density values (the stars) in the upper panel of the figure, you can see that the rate in the inner 250 pc (the CMZ) is $\sim 10^{2.5} \approx 300$ solar masses per Gyr per pc$^{2}$, while at a radius of 5 kpc, the rate is only $\sim 10^{0.6} \approx 4$. But the area of a 1-kiloparsec-wide annulus at $R = 5$ kpc is $\sim 3 \times 10^{7}$ pc$^{2}$, while the area of the CMZ is only $\sim 2 \times 10^{5}$ pc$^{2}$. So even this one annulus within the disk has about twice the star formation rate (solar masses per Gyr) of the CMZ. Add this up for the whole disk, and you do indeed get most of the star formation happening in the disk, not the center.
A: Stellar formation requires that gas clouds become unstable against gravitational contraction. This in turn requires that their gravitational potential energy exceeds their thermal energy (the Jeans instability). But in the galactic core there is a lot of radiation heating the gas, making it too hot to coalesce. In fact, the radiation is likely to push the gas outwards. In the disk there are more opportunities to cool down.
