Would there be any advantages to a wing with a concave upper surface?
The idea behind this design seems to be that the principal cause of lift
of a wing is that the top has a larger area than the bottom, so if we
substitute a top surface with a different shape but the same area, we will
still get similar lift.

This is the idea behind the equal-transit-time theory of lift,
which is incorrect, disagreeing in several ways with
with actual observations of air flowing past airfoils and the
lift forces that result.
NASA has produced a basic but thorough refutation of this theory;
among the points they make are


  The lift predicted by the "Equal Transit" theory is much less than the observed lift, because the velocity is too low. The actual velocity over the top of an airfoil is much faster than that predicted by the "Longer Path" theory and particles moving over the top arrive at the trailing edge before particles moving under the airfoil.


and


  There are modern, low-drag airfoils which produce lift on which the bottom surface is actually longer than the top.


What you generally want in an airfoil is for it to establish streamlines
that carry the air at very high speed over the top and shoot it downward
past the trailing edge. The speed is not determined by the surface area
or length along the top surface; the speed is determined by other things,
and two air molecules that start near each other but go on different sides
of the wing (one on top, one on the bottom) get to the trailing edge when their respective speeds carry them there, generally not at the same time as each other.

Putting a big hollow space on top of the wing
where the air can pile up and then forcing it over a second hump to get
to the trailing edge seems counterproductive.
It might just lead to the wing stalling at relatively low angles of attack,
which would limit the amount of lift you can get from it.
But I am not an airfoil designer, and I don't know exactly what the result would be.