This note discusses the recent work of Huang-Schoen-Wang [13] on specifying total conserved quantities of a vacuum initial data set.

A version of the Donaldson-Thomas invariants of a Calabi-Yau threefold is proposed as a conjectural mathematical definition of the Gopakumar-Vafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a new viewpoint on the computation of the local Gromov-Witten invariants of contractible curves by Bryan, Leung, and the author.

Building on [BB1] we prove a general criterion for convergence of (possibly singular) Bergman measures towards pluripotential-theoretic equilibrium measures on complex manifolds. The criterion may be formulated in terms of the growth properties of the unit-balls of certain norms on holomorphic sections, or equivalently as an asymptotic minimization property for generalized Donaldson$L$-functionals. Our result settles in particular a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points and it gives the convergence of Bergman measures towards the equilibrium measure for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.

In this paper, we show that any compact K\"ahler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K\"ahler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:\C\> X$.

The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmuller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra.