Günter M. Ziegler: Constructing surfaces, combinatorial and geometric

Constructions by Heffter (1891), Ringel (1962), and by Datta (2005) yield combinatorial schemes for interesting/extremal closed polyhedral surfaces. However, it is a long way from the combinatorial description via topological classification to geometric realizations of such surfaces. In this lecture, we will in particular present the (surprisingly simple and explicit) surfaces by Datta, ask lots of questions about them, and connect them to a number of challenging open problems about the maximal "complexity" of polyhedral surfaces in 3-space. We then go on and connect this with the maximal complexity of 3-dimensional polyhedral tilings, and thus with the "fatness" parameter of 4-dimensional polytopes.