Exact Results in Quantum Theory

I will present an introductory talk about the geometry of the universal covering of SL(2, R). I will first present SL(2, R) and I will show how this group can be interpreted as a surface. Then I will continue to the description of hyperbolas in 1 and 2 dimensions with the aim to give a general overview of how to think about elements of the group SL(2, R) which can be seen as points on a 3 dimensional hyperbola (or anti-de Sitter (AdS) space). I will present the algebra of SL(2, R) group and one parameter subgroups generated by them in SL(2, R). Then after introducing the KAN decomposition I will present how the idea of the universal cover emerges from it. I will conclude with presenting how the universal covering group can be imagined with the use of concepts previously introduced.