Abstract: This talk is an introduction to the classical dynamical systems and the way it can be generalized to the quantum setting. We shall start with a Hamiltonian on the phase space and show that how the flow can be lifted on the observables after quantization. The Heisenberg equation can be obtained from its classical counterpart, i.e. Hamilton's equations. It is a well known fact that the geodesic flow of a Riemannian manifold $M$ is the flow of the Hamiltonian given by $H(q,p)=g_q(p,p)/2$. This can be generalized to spectral triples. We also show that for a finitely summable (even) spectral triple $(A,H,D)$ the analogue of the geodesic flow on the bounded operators of H, is given by $$F_t(T)=e^{it|D|}T e^{-it|D|}.$$ Finally I shall recall Egorov's theorem.

This is the first talk in a series of talks on "quantum dynamical systems and their properties" which will be jointly delivered by Ali Fathi and Asghar Ghorbanpour.

Abstract: The rational homotopy type $X_{\mathbb{Q}}$ of an arbitrary space $X$ has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space $X$ are not accessible through the space $X_{\mathbb{Q}}$. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces Toen introduced the notion of a pointed schematic homotopy type over a field $\mathbb{k}$, $(X\times k)^{sch}.$

In his recent study of the pro-nilpotent Grothendieck-Teichmuller group via operads, Fresse makes use of the rational homotopy type of the little $2$-disks operad $E_2$. As a first step in the extension of Fresse's program to the pro-algebraic case we discuss the existence of a schematization of the little $2$-disks operad.