We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a $4\times 4$ Riemann–Hilbert problem related to the Hastings–McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in noncolliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes like a square root at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.

Strange duality is shown to hold over generic $K3$ surfaces in a large number of cases. The isomorphism for elliptic $K3$ surfaces is established first via Fourier–Mukai techniques. Applications to Brill–Noether theory for sheaves on $K3$ surfaces are also obtained. The appendix, written by Kota Yoshioka, discusses the behavior of the moduli spaces under change of polarization, as needed in the argument.

In this paper we prove that if $\varphi :\mathbb{C}\to \mathbb{C}$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \mathbb{C}$ is a compact set contained in a ball $B$, then

$$\frac{{Ċ}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}\left(E\right)}{diam(B{)}^{\frac{2}{K+1}}}\ge c^{-1}(\frac{\gamma \left(\varphi \right(E\left)\right)}{diam\left(\varphi \right(B\left)\right)}{)}^{\frac{2K}{K+1}},$$

where $\gamma$ stands for the analytic capacity and ${Ċ}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a nonlinear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e., removable for bounded $K$-quasiregular maps), it has positive capacity ${Ċ}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$. This improves previous results that assert that $E$ must have non-$\sigma$-finite Hausdorff measure of dimension $\frac{2}{K+1}$. We also show that the indices $\frac{2K}{2K+1}$, $\frac{2K+1}{K+1}$ are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are $K$-removable. So essentially we solve the problem of finding sharp “metric” conditions for $K$-removability.