We prove a version of the Arezzo-Pacard-Singer blow-up theorem in the setting of Poincar\'e type metrics. We apply this to give new examples of extremal Poincar\'e type metrics. A key feature is an additional obstruction which has no analogue in the compact case. This condition is conjecturally related to ensuring the metrics remain of Poincar\'e type.

Locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^2$ were classified by E.\,Cartan in 1932. In this work, we complete the classification of locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$.

We give a definition for the restriction of a difference module on the affine line to a formal neighborhood of an orbit, trying to mimic the analogous definition and properties for a D-module. We show that this definition is reasonable in two ways. First, we show that specifying a difference module on the affine line is equivalent to giving its restriction to the complement of an orbit, together with its restriction to a neighborhood of an orbit and an isomorphism between the restriction of both to the intersection. We also give a definition for vanishing cycles of a difference module and define a local Mellin transform, which is an equivalence between vanishing cycles of a difference module and nearby cycles of its Mellin transform, a D-module.

We classify all links whose Khovanov homology have ranks no greater than 8, and all three-component links whose Khovanov homology have ranks no greater than 12, where the coefficient ring is Z/2. The classification is based on the previous results of Kronheimer-Mrowka, Batson-Seed, Baldwin-Sivek, and the authors.

We construct, for every prime p, a function field K of characteristic p and an ordinary abelian variety A over K, with no isotrivial factors, that admits an etale self-isogeny of p-power degree. As a consequence, we deduce that there exist ordinary abelian varieties over function fields whose groups of points over the maximal purely inseparable extension is not finitely generated, answering in the negative a question of Thomas Scanlon.

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