We utilize recent results of André and Gabber on the existence of weakly functorial, integral perfectoid big Cohen–Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed characteristic BCM-variant of rational/F-rational singularities, of log terminal/F-regular singularities, and of multiplier/test ideals of divisor pairs. We prove a number of results about these objects including a restriction theorem for perfectoid BCM multiplier/test ideals and deformation statements for perfectoid BCM-regular and BCM-rational singularities. As an application, we obtain results on the behavior of F-regular and F-rational singularities in arithmetic families.

Suppose that M is a complete, embedded minimal surface in ${\mathbb{R}}^3$ with an infinite number of ends, finite genus, and compact boundary. We prove that the simple limit ends of M have properly embedded representatives with compact boundary, genus zero, and constrained geometry. We use this result to show that if M has at least two simple limit ends, then M has exactly two simple limit ends. Furthermore, we demonstrate that M is properly embedded in ${\mathbb{R}}^3$ if and only if M has at most two limit ends if and only if M has a countable number of limit ends.

We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. For the 2D Euler equation in the patch setting, we show that every uniformly rotating patch D with angular velocity $\mathrm{\Omega }\le 0$ or $\mathrm{\Omega }\ge \frac{1}{2}$ must be radial, where both bounds are sharp. For the gSQG equation, we obtain a similar symmetry result for $\mathrm{\Omega }\le 0$ or $\mathrm{\Omega }\ge {\mathrm{\Omega }}_{\mathit{\alpha }}$ (with the bounds being sharp), under the additional assumption that the patch is simply connected. These results settle several open questions posed by Hmidi, de la Hoz, Hassainia, and Mateu on uniformly rotating patches. Along the way, we close a question by Choksi, Neumayer, and Topaloglu on overdetermined problems for the fractional Laplacian, which may be of independent interest. The main new ideas come from a calculus-of-variations point of view.