We prove that the 2-Deligne tensor product of two compact semisimple 2-categories exists. Further, under suitable hypotheses, we explain how to describe the $\mathrm{Hom}$-categories, connected components, and simple objects of a 2-Deligne tensor product. Finally, we prove that the 2-Deligne tensor product of two compact semisimple tensor 2-categories is a compact semisimple tensor 2-category.

In this paper, we introduce a strongly quasi-local version of the coarse Novikov conjecture, which states that a certain assembly map from the coarse K-homology of a metric space to the K-theory of its strongly quasi-local algebra is injective. We prove that the conjecture holds for metric spaces with bounded geometry which can be coarsely embedded into Banach spaces with Property (H), as introduced by Kasparov and Yu. We also generalize the notion of strong quasi-locality to proper metric spaces and provide a (strongly) quasi-local picture for K-homology.

We consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to $K3$ surfaces and Fano threefolds. In particular, we consider $K3$ surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pullback of plane curves of degree $k⩾7$ lies on a unique $K3$ surface. If $k⩽6$, then the general such curve is instead extendable to a higher dimensional variety. In the cases $k=4,5,6$, this gives the existence of singular index-k Fano varieties of dimensions 8, 5, 3, genera 17, 26, 37, and indices 6, 3, 1, respectively. For $k=6$, we recover the Fano variety $\mathbf{P}(3,1,1,1)$, one of only two Fano threefolds with canonical Gorenstein singularities with the maximal genus 37 found by Prokhorov. We show that the latter variety is not further extendable. For $k=4$ and 5, these Fano varieties have been identified by Totaro. We also study the extensions of smooth degree-2 sections of $K3$ surfaces of genus 3. In all these cases, we compute the corank of the Gauss–Wahl maps of the curves under consideration. Finally, we observe that linear systems on double covers of the projective plane provide superabundant logarithmic Severi varieties.

Up to a finite étale covering, we classify a smooth projective 3-fold X with $\mathrm{\kappa }(X)=-\mathrm{\infty }$ admitting a nonisomorphic étale endomorphism. We mainly study the case where there exist an FESP $Y_{\bullet }$ constructed from X by a sequence of blowing-downs of an ESP and an extremal ray $R_{\bullet }$ of $\overline{NE}(Y_{\bullet })$ such that the contraction morphisms associated to $R_{\bullet }$ give $Y_{\bullet }$ a smooth del Pezzo fiber space over an elliptic curve.

We present q-Bernoulli and q-Euler polynomials generated by the third Jackson q-Bessel function. We derive new q-analogues of the Lidstone expansion theorem. First, we expand a certain class of entire functions in terms of the q-Bernoulli polynomials we introduce. The coefficients of the polynomials are the even powers of the symmetric q-derivative $\frac{{\mathit{\delta }}_{q}f(z)}{{\mathit{\delta }}_{q}z}$ at 0 and 1. The other forms expand a certain class of functions in the q-Euler polynomials, where the coefficients of the expansions are the even and odd powers of the symmetric q-derivative $\frac{{\mathit{\delta }}_{q}f(z)}{{\mathit{\delta }}_{q}z}$ at 0 and 1.

For a finitely generated module M over a commutative Noetherian ring R, we settle the Auslander–Reiten conjecture when at least one of ${Hom}_R(M,R)$ and ${Hom}_R(M,M)$ has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.

Let D be a division ring with center F, and let G be an almost subnormal subgroup of $D^{\ast }$. In this paper, we show that if G contains a non-abelian locally solvable maximal subgroup, then D must be a cyclic algebra of prime degree over F. Moreover, it is proved that every locally nilpotent maximal subgroup of G is abelian.