01143nas a2200121   4500008004100000245005700041210005700098260001000155520076800165653002800933100002400961856003600985       2007                        en  d00aNoncommutative geometry and quantum group symmetries0 aNoncommutative geometry and quantum group symmetries  bSISSA3 aIt is a widespread belief that mathematics originates from the desire to understand (and eventually to formalize) some aspects of the real world. Quoting [Man07],   «we are doing mathematics in order to understand, create, and handle things, and perhaps this understanding is mathematics»  . Let me thus begin with a brief discussion of the physical ideas that motivated the development of Noncommutative Geometry and Quantum Group Theory - the areas of mathematics to which this dissertation belongs. Some physicists believe, and Einstein himself expressed this view in [Ein98a], that physics progresses in stages: there is no `final\\\' theory of Nature, but simply a sequence of theories which provide more and more accurate descriptions of the real world...10aNoncommutative geometry1 aD'Andrea, Francesco  uhttp://hdl.handle.net/1963/5269