I am going to prove that in many cases the rational cohomology ring of groups of symplectomorphisms is infinitely generated (as a ring!). The first part of the talk will be concerned with algebraic topology methods (rational homotopy, Gottlieb groups etc). The main result will appear as the (not so direct) application. Details can be found in my recent paper Evaluation fibrations and topology of symplectomorphisms (math.AT/0305325). To understand the talk it is enough to know basic algebraic topology.