Local Systems, Positivity and (Non-Commutative) Markov Numbers
Markov numbers are positive integers that appear in solutions of the equation x^2+y^2+z^2=3xyz. They also appear naturally when one tries to parametrize positive $SL_2(R)$-local systems on a one punctured torus. In this talk I will explain that there are generalizations of such positive local systems for more general semisimple Lie groups. I will then discuss a special case which is related to groups of type $SL_2$ over non-commutative rings and describe the positive structures associated to this. In the end I will show how this leads to non-commutative deformations of Markov numbers.