will explore the role that conformal symmetries may play in cosmology. First,
we will discuss the symmetries underlying the statistics of the
primordial perturbations which seeded the temperature anisotropies of the
Cosmic Microwave Background. I will show how symmetry considerations lead us to
three broad classes of theories to explain these perturbations: single-field
inflation, multi-field inflation, and the conformal mechanism. We will discuss
the symmetries in each case and derive their model-independent consequences.
Finally, we will examine the possibility of violating the null energy condition
with a well-behaved quantum field theory.