Title: Nonperturbative quantum and conformal gravity

Abstract: Loop Quantum Gravity (LQG) and Spin Foam (SF) models are an 
attempt at a diffeomorphism invariant or non-perturbative formulation 
of quantum gravity. In the first part of this talk we will present an 
introduction to LQG and the SF models beginning by a brief motivation, 
reviewing the method of Dirac constraint quantization and applying it 
to the Holst action, thus obtaining LQG. We will then quickly skim over 
the main results that this formulation of quantum gravity has obtained 
so far, some of its most important open questions and introduce the SF 
models that are an attempt at a path integral formulation of LQG. 
We shall also review the main results of the SF approach as well as 
some of its main open questions.

In the second half of the talk, we will look more in detail at the 
twistor formulation of the SF models. Especially introducing the 
twisted geometry of the spinnetworks, a 3d quantum polyhedral geometry 
with a discontinuous metric. Spin networks are the states between which 
the transition amplitudes are calculated in the SF approach to quantum 
gravity and the kinematical states of the LQG approach. We shall then 
motivate the interest in conformally invariant theories and sketch how 
the spinnetworks should be generalizable to this setting 
(a work in progress).