An introduction to noncommutative geometry and noncommutative tori

"Noncommutative geometry" in the sense of Alain Connes and others is the study of geometry on "spaces" where the "algebra of functions" is noncommutative. Such spaces naturally come up in representation theory and physics. The simplest nontrivial example of such a space is the "noncommtutative 2-torus". We will give a thorough introduction to this space and discuss some of the interesting problems it presents.