School of Natural Sciences
We analyze modular invariance drawing inspiration from tauberian
theorems. Given a modular invariant partition function with a positive
spectral density, we derive lower and upper bounds on the number of
operators within a given energy interval. They are most revealing at high
energies. In this limit we rigorously derive the Cardy formula for the
microcanonical entropy together with optimal error estimates for various
widths of the averaging energy shell. Finally, we identify a new universal
I will discuss time-dependent probes of the renormalization group, and derive new constraints that govern the spread of local operators in holographic theories. The same methods lead to sum rules for inflationary correlators, relating observables, like the speed of sound during inflation, to properties of the UV theory.
Several lines of evidence, both theoretical and observational, indicate that additional planets in the outer solar system remain to be discovered. Most recently, attention has been focused on 'Planet Nine',a few Earth mass planet with a moderate inclination at a distance of a few hundred AU (Trujillo & Sheppard 2014, Batygin & Brown 2016, Brown & Batygin 2016). I will review this evidence for the existence of this body and will present a variety of approaches that are being used to find it.