Some results on Tilting Theory

We describe some more or less new results on Tilting Theory, concerning both discrete objects and continuous objects. On th one hand, almost all useful discrete objects presented in the lecture (that is modules, bimodules and complexes with a tilting or cotilting-type behavior) are actually vector spaces over some field, or more generally modules over some ring. An important property of these modules is their rigid structure, more precisely the property of being without selfextensions. On the other hand,also also continuos objects (that is more or less visible classes of modules, described by means of functors and exact sequences) may be often described in a combinatorial way. For instance, in case of modules over algebras of finite representation type, we may decide whether or not two classes of modules have similar or opposite properties (that is are related to each other by an equivalence or a duality), by simply looking at the shape of their Auslander - Reiten quiver, an oriented graph with finitely many vertices and arrows. As pointed out in the announcement of the Tilting Tagung Twenty years of Tilting Theory, an Interdisciplinary Symposium (november 2002 , Fraueninseln, Germany), finite dimensional dimensional algebras are the context where tilting modules were born (www. mathematik.uni-muenchen.de) We also recall that a name coming fromPhysics, namely the word reflection appears in the title of the famous paper Generalizations of the Bernstein - Gelfand - Ponomarev reflection functors (Springer LMN 832, 1980), where S. Brenner and M.C.R. Butler introduced tilting modules for the first time.