Title: Progress on Turán's brick factory problem

Consider the problem of drawing a complete bipartite graph K_n,m in the plane such that as few edges as possible intersect. The minimum number of intersections for all drawings in the plane is called the crossing number of K_n,m. The problem of determining the crossing number of K_n,m has been open for decades, and has only been settled for the case where min(m,n) < 7. It sometimes called Turán's brick factory problem.
In this talk we will describe some recent progress that can be obtained by relaxing the problem to a (non-covex) quadratic optimization problem over the simplex, and then approximating the latter problem using semidefinite programming.
In particular, we will present improved lower bounds on the crossing number of K_7,m, and discuss the implications.
Joint work with Dima Pasechnik, Bruce Richter and Gelasio Salazar.