Abstract: After introducing the notion of a quantum dynamical system and ergodicity of such systems, I will explain the example of the quantum Kronecker map on noncommutative 2-torus. If time allows I will also explain the notion of semi-classical ergodicity (quantum ergodicity) that arises in quantization of chaotic Hamiltonian systems.

Abstract: A comparison of the etale homotopy type of a representable geometrically pointed simplicial sheaf X and that of its standard and twisted fibred sites is demonstrated. This result is applied to yield an etale van Kampen theorem for representable geometrically pointed connected simplicial sheaves under suitable hypotheses on the ambient site.