I will discuss some problems arising in the study of toric Kaehler metrics, mostly focusing on studying the invariant spectrum of the Laplacian, explicit constructions of distinguished metrics (Einstein, Ricci soliton, and quasi-Einstein metrics) and connections between these topics. Time permitting, I will also outline numerical approaches to these problems.

I will discuss a Kuranishi-type theorem for deformations of complex structure on ALE Kahler surfaces, which will be used to prove that for any scalar-flat Kahler ALE surface, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics can then be constructed, which is universal up to small diffeomorphisms. I will also discuss a formula for the dimension of the local moduli space in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of an isolated quotient singularity.  This is joint work with Jiyuan Han.

We explore the structure of the singularities of Yang-Mills flow in dimensions n ≥ 4. First we derive a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow at such singular points, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set. This is joint work with Jeffrey Streets

An oriented hypersurface in a hyperkaehler 4-manifold naturally inherits a coclosed coframing.  Bryant showed that, in the real analytic case, any oriented 3-manifold with a coclosed coframing can always be locally “thickened” to a hyperkaehler 4-manifold, in an essentially unique way.  This raises the natural question: when can these 3-manifolds with this structure arise as the boundary of a hyperkaehler 4-manifold?  In particular, starting from a compact hyperkaehler 4-manifold with boundary, which deformations of the boundary structure can be extended to a hyperkaehler deformation of the interior?  I will discuss recent progress on this problem, which is joint work with Joel Fine and Michael Singer.

We consider the evolution by mean curvature flow of surface clusters,
where along triple edges three surfaces are allowed to meet under an equal angle
condition. We show that any such smooth flow, which is weakly close to the static
flow consisting of three half-planes meeting along the common boundary, is smoothly
close with estimates. Furthermore, we show how this can be used to prove a smooth
short-time existence result. This is joint work with B. White.