Don't Lie: a gentle introduction to Lie groups

Lie groups are one of the most fundamental tool in mathematics. They are ubiquitous, starting from mathematical physics, passing through number theory and landing into the world of dynamical systems. In this talk I will give a general overview about the definition of Lie group and I will list some of its property. My attention will be mainly focused on two aspects. I am going to talk about the Iwasawa decomposition of non-compact Lie groups, which can be interpretated as a suitable generalization of QR-factorization. This decomposition result will allow to study some topological properties of Lie groups. Then I will move to some analytical aspects related to measure theory. Studying unimodular lattices on the plane we will "compute" the volume of the quotient SL(2,R)/SL(2,Z) in terms of Haar measure. This will be a very introductory seminar which will be the starting point of a minicourse about Lie groups and representations of lattices.