It is a classical result that groupoids model homotopy 1-types, in the sense that there is an equivalence between the homotopy categories, via the classifying space and fundamental groupoid functors. We extend this to stable homotopy 1-types and Picard groupoids. Using an algebraic description of Picard groupoids, we give a model for the Postnikov invariant of a stable 1-type and describe the action of the truncated sphere spectrum in these terms. We relate this data to exact sequences of Picard groupoids developed by Vitale, constructing a model for the homotopy cofiber of a map of stable 1-types. Joint with Angélica Osorno.

In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study the equations of motion of a rigid body in R^3 and the equations of ideal hydrodynamics. I will describe how to extend his formalism and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bi-Hamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and global existence of solutions.

Introduced by R.T. Powers, $E_0$-semigroups are one-parameter semigroups $\rho=\{\rho_t\}_{t\geq 0}$ of unital normal *-endomorphisms acting on von Neumann algebras, usually the von Neumann algebra $B(H)$ of all bounded linear operators on a separable Hilbert space $H$. $E_0$-semigroups can be regarded as quantum generalizations of the classical time-irreversible dynamical systems, and their study takes into account at a non-commutative level various dissipation mechanisms and state-time evolution phenomena.

It is our purpose, in this presentation, to survey the current state of knowledge of the subject, and to discuss several classification problems.