Let F(X_0,...,X_{n+1}) be a homogeneous polynomial of degree d. When do there exist and integer r > 0 and rd linear forms L_{ij} such that F^r = det [L_{ij}]? This is a very classical question investigated starting in 1855. It is nowadays shown that it is equivalent to the existence of an Ulrich sheaf on the hypersurface $X=V(F) subset P^{n+1}.

In the talk we will introduce Ulrich bundles and briefly talk about their importance, the main conjecture being that any variety should carry one. We will then focus on "geometrizing" the problem, namely, to show that the existence of an Ulrich bundle is equivalent, in a Hartshorne-Serre-like correspondence, to the existence of a certain subvariety. We will apply this to prove that complete intersections do not have low rank Ulrich bundles. 

https://www.matem.unam.mx/~lozano/eseminar.html