Abstract: This talk will provide an overview of the renormalized volume
coefficients and associated renormalized volume functionals in conformal
geometry. These are Riemannian invariants constructed from the volume
expansion of a Poincar\'e-Einstein metric with a prescribed conformal
infinity. They arose in the context of the AdS/CFT correspondence in
physics. They have some surprising properties, which, among other things,
suggests a natural variant of the so-called $\sigma_k$-Yamabe problem, a
fully nonlinear generalization of the Yamabe problem which has been the
focus of much attention during the last decade.

In this talk I will introduce a natural elliptic equation (the
vortex equation) in two dimensional gauged linear sigma-model. It
generalizes the Cauchy-Riemann equation in Gromov-Witten theory, and the
Hermitian-Einstein equation in the theory of vector bundles. I will also
discuss the "large-area limit" of the vortex equation and its relation with
the nonlinear sigma-model. Almost everything will be discussed in the
context of line bundles over Riemann surfaces.

Recently Streets and Tian introduced a geometric flow of almost-Hermitian
structures. We will discuss the motivation for considering such a flow. Moreover, we
will give evidence that the flow reflects the underlying almost-Hermitian structure
of almost complex manifolds.

Advisor: Abel Klein