I will discuss prospects of formulating simple extensions of the standard model and gravity that exhibit local Weyl symmetry in the ultraviolet. The principal advantage of such constructions is that they naturally address the gauge and gravitational hierarchy problem. Furthermore,  I will argue that Cartan-Einstein gravity provides a natural framework for conformal symmetry, as this theory contains torsion vector which can be interpreted as the Weyl vector.

We show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form.

In this talk I will discuss how a gravitationally collapsing object emits classical radiation at a rate equal to the quantum radiation rate if suitable initial conditions are chosen for two classical fields. The coupled dynamics of gravitational collapse and radiation are then solved for in a simple toy model, illustrating how the classical system may be used to gain insight into Hawking evaporation.

There is experimental evidence that the quark-gluon plasma produced in ultra-relativistic heavy ion collisions is well described by viscous hydrodynamics, with a low value of the viscosity (relative to the entropy density). This observation, added to the recent discovery that the same description works well also for high energy proton-nucleus or high multiplicity proton-proton collisions, is raising a number of interesting theoretical questions:  How does the system of gluons freed in the early stage of a collision evolve towards local thermal equilibrium?

Turbulence is an ubiquitous phenomenon in natural and industrial fluid flows. Yet, it still lacks a  satisfactory theoretical description. One of the main open issues is to calculate the statistical properties of the turbulent steady state, and in particular what is generically called intermittency effects, starting from the fundamental description of the fluid dynamics provided by Navier-Stokes equation. In this presentation, I will focus on isotropic and homogeneous turbulence in three-dimensional incompressible flows.