The crossing number of complete bipartite graphs

We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph $K_{m,n}$ is $\lfloor \frac{m}{2}\rfloor \lfloor \frac{m-1}{2}\rfloor \lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor .$