Consider the life of a single dust particle that randomly gets kicked up into the air, at which point it floats around for a while before being deposited on the floor again. One (very) simple model of this process might be as follows:
1) the probability of the particle getting kicked up in 1 second increases with the airspeed at it's location, and
2) once it's in the air, where it falls is random.
Then one would expect the particle to spend more time on the ground where the airspeed is low, and less where the airspeed is high, therefore accumulating in low-speed locations. If this model is legit, the question becomes "why does the air move slower in corners?"
Empirically, (at least from walking around my apartment) it seems that the airflow is indeed slower in the corners, so this is a good sanity-check.
To really do this justice, you'd have to consider the full Navier-stokes equations, but those are hard, so lets consider a simplified model. Suppose the flow is an incompressible, slow, pressure-driven flow with a few sources and sinks (air conditioner, cracks under the door, etc), and no-flow boundary conditions at the walls. In this case, the pressure field should solve the Laplace equation, which we have some intuition about it as some sort of smooth rubber surface trying to be "as flat as possible".
The no-flow condition means that the normal derivative of the pressure at a wall is zero. Now consider a corner between one wall along x=0, and another wall along y=0. If we are near enough to the corner, we expect to be within the realm of influence of both walls, so we expect both derivatives dP/dx and dP/dy to be small, since they are zero at the y=0 and x=0 walls respectively. Therefore the pressure gradient is small, so the velocity is small in the corner, as expected.
Summary: a simple particle model predicts dust will accumulate where the air is moving slowly, and a simple pressure-driven flow model predicts flow to be slow near corners. Whether these models are legit or not is worthy of debate, but they do predict the right behavior in this case.