In this talk We discuss the Martin compactification of a special complete noncompact
surface with negative Gaussian curvature which arises in our study of infinitesimal
rigidity of three-dimensional (collapsed) steady gradient Ricci solitons. In
particular, we investigate positive eigenfunctions with eigenvalue one of the
Laplace operator and prove a uniqueness result: such eigenfunctions are unique up to
a positive constant multiple if certain boundary behavior is satisfied. This
uniqueness result was used to prove an infinitesimal rigidity theorem for
deformations of certain three-dimensional collapsed gradient steady Ricci soliton
with a non-trivial Killing vector field. It is a joint work with Huai-Dong Cao.

We continue the discussion of Viale-Weiss paper ``On the consistency strength of the proper forcing axiom". We complete the proof that PFA implies existence of stationarily many guessing models.

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.