In figure 3 of this document, there is data relating $\Re(\sigma(\omega))$ to the Fermi energy. It is claimed that $\Re(\sigma(\omega))$ is determined via reflectivity measurements. How is this done? What is the formula relating the two?
The optical conductivity $\sigma$ is basically equivalent to the dielectric function $\epsilon$:
$$ \sigma(\omega) = i\omega\epsilon_0 (1-\epsilon(\omega)) $$
So the real part of the conductivity contains the same information as the imaginary part of the dielectric function:
$$ \sigma'(\omega) = \epsilon_0 \epsilon''(\omega) \omega $$
You can determine the dielectric function from the reflectance; it is the square root of the complex index of refraction, which you can determine from angle-dependent reflectance measurements for s and p polarization (basically, ellipsometry.)