PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.  The purpose of my talk will be to present some new results for the Littlewood and Mixed Littlewood conjectures. In the first part of the discussion I will give an expository overview of what is currently known about the two conjectures and present in more detail a particular problem connected to them. In the second part I will give an overview of the arguments that go into proving the problem for the mixed case.