Kristian Ranestad (UiO) - Smooth surfaces not of general type in P^4, old and new

Since Ellingsrud and Peskine showed that there is an upper bound on the degree of smooth surfaces not of general type in P^4, there have been efforts to find them. I will give a survey of different methods of construction, and a list of the kind of surfaces known so far, reporting also on joint work with Hirotachi Abo and Frank-Olaf Schreyer to write a Macaulay2 package that produces equations of examples of each kind.