Spaces of matrices of constant rank and instantons.

Given a complex vector space V of dimension n, one can look at d-dimensional linear subspaces A in \Wedge^2(V), whose elements have constant rank r. The natural interpretation of A as a vector bundle map yields some restrictions on the values that r,n and d can attain.
After a brief overview of the subject and of the main techniques used, I will concentrate on the case r=n-2 and d=4. I will introduce what used to be the only known example, by Westwick, and give an explanation of this example in terms of instanton bundles and the derived category of P^3. I will then present a new method that allows to prove the existence of new examples of such spaces, and show how this method applies to instanton bundles of charge 2 and 4. These results are in collaboration with D.Faenzi and E.Mezzetti.