For any positive integer n, we give a ${\mathbb{Z}}^{\mathit{n}}$-cork with a ${\mathbb{Z}}^{\mathit{n}}$-effective embedding in a 4-manifold being homeomorphic to $\mathit{E}(\mathit{n})$. This means that a cork gives a subset ${\mathbb{Z}}^{\mathit{n}}$ in the differential structures on $\mathit{E}(\mathit{n})$. Further, we describe handle decompositions of the twisted doubles (homotopy ${\mathit{S}}^4$) of Gompf’s infinite order cork and show that they are log transforms of ${\mathit{S}}^4$.

We show that the Dehn function of the handlebody group is exponential in any genus $g\ge 3$. On the other hand, we show that the handlebody group of genus 2 is cubical and biautomatic and therefore has a quadratic Dehn function.

We consider a compact CR manifold with a transversal CR locally free circle action endowed with an ${\mathit{S}}^1$-equivariant positive CR line bundle. We prove that a certain weighted Fourier–Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.

If X is a smooth hypersurface in complex projective space, the Fano variety of lines on X is stratified by the splitting type of the normal bundle of the line. We show that for general hypersurfaces, these strata have the expected dimension and, in this case, compute the class of the closure of the strata in the Chow ring of the Grassmannian of lines in projective space. For certain splitting types, we also provide upper bounds on the dimension of the strata that hold for all smooth X.

We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups. This allows us to produce new nonarithmetic lattices, bringing the number of known nonarithmetic commensurability classes in $\mathrm{PU}(2,1)$ to 22.

In this short note, we compare the combinatorial sign assignment of Manolescu, Ozsváth, Szabó, and Thurston for grid homology of knots and links in ${\mathit{S}}^3$ with the sign assignment coming from a coherent system of orientations on Whitney disks. Although these constructions produce different signs, a small modification of the convention in either of the two methods results in identical sign assignments, and thus identical chain complexes.

We derive asymptotics for the scaling of the total diffraction intensity for the set of k-free integers near the origin, which is a measure for the degree of patch fluctuations.