We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, as defined by Mann and Rafi, and with 0 or infinite genus, we show that when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.

Inspired by a conjecture of Vladimir Maz’ya on Φ-inequalities in the spirit of Bourgain and Brezis, we establish some Φ-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of Φ-inequalities for differential operators. The proofs rest on new simple Bellman functions.

For a finitely generated group G, let $\mathit{H}(\mathit{G})$ denote Bowditch’s taut loop length spectrum. We prove that if $\mathit{G}=(\mathit{A}\ast \mathit{B})/⟨⟨\mathcal{R}⟩⟩$ is a ${\mathit{C}}^{\prime }(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $\mathit{H}(\mathit{G})$ is equivalent to $\mathit{H}(\mathit{A})\cup \mathit{H}(\mathit{B})$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch’s construction of continuously many non-quasi-isometric ${\mathit{C}}^{\prime }(1/6)$ small cancellation 2-generated groups to obtain our main result: Let $\mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups:

(1) $\{\mathit{G}\in \mathcal{G}:\underset{\_}{\mathrm{cd}}(\mathit{G})=2\text{and}\underset{\_}{\mathrm{gd}}(\mathit{G})=3\}$;

(2) $\{\mathit{G}\in \mathcal{G}:\underset{\_}{\underset{\_}{\mathrm{cd}}}(\mathit{G})=2\text{and}\underset{\_}{\underset{\_}{\mathrm{gd}}}(\mathit{G})=3\}$;

(3) $\{\mathit{G}\in \mathcal{G}:{\mathrm{cd}}_{\mathbb{Q}}(\mathit{G})=2\text{and}{\mathrm{cd}}_{\mathbb{Z}}(\mathit{G})=3\}$.

On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented ${\mathit{C}}^{\prime }(1/12)$ small cancellation quotients of free products; in particular, this produces new examples of groups exhibiting an Eilenberg–Ganea phenomenon for families.

We also show that if there is a finitely presented counterexample to the Eilenberg–Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counterexamples.

We establish a congruence satisfied by the integer group determinants for the non-Abelian Heisenberg group of order ${\mathit{p}}^3$. We characterize all determinant values coprime to p, give sharp divisibility conditions for multiples of p, and determine all values when $\mathit{p}=3$. We also provide new sharp conditions on the power of p dividing the group determinants for ${\mathbb{Z}}_{\mathit{p}}^2$.

For a finite group, the integer group determinants can be understood as corresponding to Lind’s generalization of the Mahler measure. We speculate on the Lind–Mahler measure for the discrete Heisenberg group and for two other infinite non-Abelian groups arising from symmetries of the plane and 3-space.

We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on p-forms by computing the heat invariants associated with the p-spectrum. We show that the heat invariants of the 0-spectrum together with those of the 1-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $\le 3$. This is enough to distinguish orbifolds from manifolds for dimension $\le 3$.

Felix Klein in the course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of 60 points in ${\mathbb{P}}^3$. This configuration has appeared in various guises, perhaps most notably as the configuration of points dual to the 60 reflection planes in the group ${\mathit{G}}_{31}$ in the Shephard–Todd list.

In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated with the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities $4,2$ and 2. Secondly, Chiantini and Migliore observed in 2020 that there are nontrivial sets of points in ${\mathbb{P}}^3$ with the surprising property that their general projection to ${\mathbb{P}}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in ${\mathbb{P}}^3$, which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.

We solve direct and inverse problems for two-dimensional (quasi) canonical systems related to exponential polynomials of a specific but sufficiently general type. The approach to the inverse problem in this paper provides an interpretation of the matrices and their determinants in the classical Schur–Cohn test for polynomials in terms of Hamiltonians of canonical systems.

Two seemingly different properties of 2-complexes were developed concurrently as criteria for the nonpositive immersion property: ‘good stackings’ and ‘bislim structures’. We establish an equivalence between these properties by introducing a third property ‘invariant staggerings’ mediating between them.