Title: The Birational Geometry of K-Moduli Spaces

Abstract: K-stability is a rapidly developing theory that allows one to construct moduli spaces for Fano varieties. In all known examples, K-moduli spaces are uniruled, so their Kodaira dimensions are negative infinity. In this talk we will describe components of K-Moduli spaces which are birational to M_g, in particular they have maximal Kodaira dimension when g is sufficiently large. This component parameterizes certain moduli spaces of vector bundles on smooth curves, and the main difficulty is to show that these moduli spaces are K-stable. To establish this we require good understanding of their toric degenerations.