The cancellation problem asks whether $\mathit{A}[{\mathit{X}}_1,{\mathit{X}}_2,\dots ,{\mathit{X}}_{\mathit{n}}]\cong \mathit{B}[{\mathit{Y}}_1,{\mathit{Y}}_2,\dots ,{\mathit{Y}}_{\mathit{n}}]$ implies $\mathit{A}\cong \mathit{B}$. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann, and Swan steadfastness can be characterized in terms of p-seminormality, which is a variant of normality introduced by Swan. We prove that p-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that p-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.

We conjecture a structure formula for the $\mathrm{SU}(\mathit{r})$ Vafa–Witten partition function for surfaces with holomorphic 2-form. The conjecture is based on S-duality and a structure formula for the vertical contribution previously derived by the third-named author using Gholampour–Thomas’s theory of virtual degeneracy loci.

For ranks $\mathit{r}=2,3$, conjectural expressions for the partition function in terms of the theta functions of ${\mathit{A}}_{\mathit{r}-1}$, ${\mathit{A}}_{\mathit{r}-1}^{\vee }$ and Seiberg–Witten invariants were known. We conjecture new expressions for $\mathit{r}=4,5$ in terms of the theta functions of ${\mathit{A}}_{\mathit{r}-1}$, ${\mathit{A}}_{\mathit{r}-1}^{\vee }$, Seiberg–Witten invariants, and continued fractions studied by Ramanujan. The vertical part of our conjectures is proved for low virtual dimensions by calculations on nested Hilbert schemes.

The horizontal part of our conjectures gives predictions for virtual Euler characteristics of Gieseker–Maruyama moduli spaces of stable sheaves. In this case, our formulae are sums of universal functions with coefficients in Galois extensions of $\mathbb{Q}$. The universal functions, corresponding to different quantum vacua, are permuted under the action of the Galois group.

For $\mathit{r}=6,7$, we also find relations with Hauptmoduln of ${\mathrm{\Gamma }}_0(\mathit{r})$. We present K-theoretic refinements for $\mathit{r}=2,3,4$ involving weak Jacobi forms.

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.

For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdős holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishes at two different positive integers under some additional conditions.

We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham, who asked for optimal and explicit order estimates for the error term $\mathrm{\Delta }(\mathit{x}):=\mathit{\psi }(\mathit{x})-\mathit{x}$, given any zero-free region $\mathcal{D}(\mathit{\eta }):=\{\mathit{s}=\mathit{\sigma }+\mathit{i}\mathit{t}\in \mathbb{C}:\mathit{\sigma }:=\mathrm{\Re }\mathit{s}\ge 1-\mathit{\eta }(\mathit{t})\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz.

Here we consider a given system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$, the corresponding integer counting function $\mathcal{N}(\mathit{x})$, and the respective error term ${\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x}):={\mathit{\psi }}_{\mathcal{P}}(\mathit{x})-\mathit{x}$ in the PNT of Beurling, where ${\mathit{\psi }}_{\mathcal{P}}(\mathit{x})$ is the Beurling analogue of $\mathit{\psi }(\mathit{x})$. First we prove that if the Beurling zeta function ${\mathit{\zeta }}_{\mathcal{P}}$ does not vanish in $\mathcal{D}(\mathit{\eta })$, then the extension of Pintz’ result holds: $|{\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x})|\le \mathit{x}exp((1+\mathit{\epsilon }){\mathit{\omega }}_{\mathit{\eta }}(log\mathit{x}))(\mathit{x}>{\mathit{x}}_0(\mathit{\epsilon }))$, where ${\mathit{\omega }}_{\mathit{\eta }}(\mathit{y}):=\mathcal{L}(\mathit{\eta })(\mathit{y})$ is the naturally occurring conjugate function—essentially the Legendre transform—of $\mathit{\eta }(\mathit{t})$, introduced into the field by Ingham. In the second part we prove a converse: if ${\mathit{\zeta }}_{\mathcal{P}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $|{\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x})|\ge \mathit{x}exp((1-\mathit{\epsilon }){\mathit{\omega }}_{\mathit{\eta }}(log\mathit{x}))$ holds “infinitely often”. This also shows that both main results are sharp apart from the arbitrarily small $\mathit{\epsilon }>0$.

The classical results of Pintz used many facts about the Riemann zeta function. Recently we worked out a number of analogous results—including some construction of quasi-optimal integration paths, a Riemann–von Mangoldt-type formula, a Carlson-type density theorem, and a Turán-type local density theorem—for the Beurling context, too. These, together with Turán’s power sum theory, all play some indispensable role in deriving the main results of the paper.

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.

Enriched curves have been studied over algebraically closed fields by Mainò [10] and recently over general base schemes in [1]. In this paper, we study enriched curves from a logarithmic viewpoint: we give a succinct definition of the stack of rich log curves as an open substack of the stack of log curves and define an enriched curve to be a curve with a minimal rich log structure on it. This logarithmic viewpoint turns out to be a natural language for enriched structures, leading to a simple modular compactification. This modular compactification is a smooth log blowup of the stack of log curves, answering affirmatively two questions from [1]. We also generalize the concept of rich curves to r-rich curves and show similar results. We include a chapter phrasing some of the key definitions solely in the language of real tropical geometry.

Let Z be an affine algebraic variety and X be a smooth flexible variety. We develop some criteria under which Z admits a closed embedding into X. In particular, we show that if $dim\mathit{X}\ge max(2dim\mathit{Z}+1,dim\mathit{T}\mathit{Z})$ and X is isomorphic (as an algebraic variety) to a special linear group or to a symplectic group, then Z admits a closed embedding into X.

We study weak approximation and the Hilbert property for Campana points, both of importance in recent work on a Manin-type conjecture by Pieropan, Smeets, Tanimoto, and Várilly-Alvarado. We show that weak weak approximation implies the Hilbert property for Campana points, and we exploit this to exhibit Campana orbifolds whose sets of Campana points are not thin.

In this paper, we study the cohomology of vector bundles on ${\mathbb{P}}^{\mathit{n}}$ defined as kernels or cokernels of general maps ${\mathit{V}}_1\to {\mathit{V}}_2$, where ${\mathit{V}}_{\mathit{i}}$ are direct sums of line bundles or certain exceptional bundles. This problem is deeply connected with fundamental results on the ranks of multiplication maps between spaces of homogeneous polynomials. Our main result establishes an asymptotic cohomology vanishing theorem. We also characterize the stability of general Steiner bundles on ${\mathbb{P}}^{\mathit{n}}$ and give a criterion for such bundles to be ample.

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $\mathit{n}\le \mathit{N}$ and integer parts of polynomials evaluated at n. The error terms in our formulas are of various strengths depending on the Diophantine properties of the leading coefficients of these polynomials.

We answer two questions about the topology of end spaces of infinite type surfaces and the action of the mapping class group that have appeared in the literature. First, we give examples of infinite type surfaces with end spaces that are not self-similar, but a unique maximal type of end, either a singleton or a Cantor set. Secondly, we use an argument of Tsankov to show that the “local complexity” relation ≼ on end types gives an equivalence relation that agrees with the notion of being locally homeomorphic.

This article provides a deeper study of the Riesz transform commutators associated with the Neumann Laplacian operator ${\mathrm{\Delta }}_{\mathit{N}}$ on ${\mathbb{R}}^{\mathit{n}}$. Along the line of singular value estimates for Riesz transform commutators established by Janson and Wolff and Rochberg and Semmes, we establish a full range of Schatten-p class characterization for these commutators.

We show that the splinter property ascends along regular residue field extensions and along arbitrary regular maps in equal characteristic. We also study the splinter property of non-Noetherian rings, especially those related to ultrapowers, to the extent necessary for our main results.

We prove a generalized version of Evans and Griffith’s improved new intersection theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least $dim\mathit{R}-dim\mathit{R}/\mathit{I}$. This improves the bound $\mathrm{ht}\mathit{I}$ obtained by Avramov, Iyengar, and Neeman in 2018.

Given a metric space X, an analyst’s traveling salesman theorem for X gives a quantitative relationship between the length of a shortest curve containing any subset $\mathit{E}\subseteq \mathit{X}$ and a multi-scale sum measuring the “flatness” of E. The first such theorem was proven by Jones for $\mathit{X}={\mathbb{R}}^2$ and extended to $\mathit{X}={\mathbb{R}}^{\mathit{n}}$ by Okikiolu, whereas an analogous theorem was proven for Hilbert space, $\mathit{X}=\mathit{H}$, by Schul. Bishop has since shown that if one considers Jordan arcs, then the quantitative relationship given by Jones’ and Okikioulu’s results can be sharpened. This paper gives a full proof of Schul’s original necessary half of the traveling salesman theorem in Hilbert space and provides a sharpening of the theorem’s quantitative relationship when restricted to Jordan arcs analogous to Bishop’s aforementioned sharpening in ${\mathbb{R}}^{\mathit{n}}$.

A smooth projective variety Y is said to satisfy Bott vanishing if ${\mathrm{\Omega }}_{\mathit{Y}}^{\mathit{j}}\otimes \mathit{L}$ has no higher cohomology for every j and every ample line bundle L. Few examples are known to satisfy this property. Among them are toric varieties, as well as the quintic del Pezzo surface, recently shown by Totaro. Here we present a new class of varieties satisfying Bott vanishing, namely stable GIT quotients of ${({\mathbb{P}}^1)}^{\mathit{n}}$ by the action of ${\mathit{PGL}}_2$ over an algebraically closed field of characteristic zero. For this, we use the work done by Halpern-Leistner on the derived category of a GIT quotient and his version of the quantization theorem. We also see that, using similar techniques, we can recover Bott vanishing for the toric case.

We study relations between the property of being log abundant for lc pairs and the termination of log MMP with scaling. We prove that any log MMP with scaling of an ample divisor starting with a projective dlt pair contains only finitely many log abundant dlt pairs. We also discuss log MMP for slc pairs.

In this article we obtain large deviation estimates for zeros of random holomorphic sections on punctured Riemann surfaces. These estimates are then employed to yield estimates for the respective hole probabilities. A particular case of relevance that is covered by our setting is that of cusp forms on arithmetic surfaces. Most of the results we obtain also allow for reasonably general probability distributions on holomorphic sections, which underlines the universal character of these estimates. Finally, we also extend our results to the case of certain higher dimensional complete Hermitian manifolds, which are not necessarily assumed to be compact.

We show that a conjecture of Kotschick about one-forms without zeros on compact Kähler manifolds follows in the case of simple Albanese torus from a conjecture of Bobadilla and Kollár about homologically trivial fibrations. As an application, we prove Kotschick’s conjecture for compact Kähler manifolds X with ${\mathit{b}}_1(\mathit{X})\ge 2dim\mathit{X}-2$, whose Albanese torus is simple.

A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the algebraic structure. In this paper we introduce a more general perspective on total symmetry and formulate a notion of “irreducibility” for totally symmetric sets in the general linear group. We classify irreducible totally symmetric sets, as well as those of maximal cardinality.

We study the Mori Dream Space (MDS) property for blowups of weighted projective planes at a general point and, more generally, blowups of toric surfaces defined by a rational plane triangle. The birational geometry of these varieties is largely governed by the existence of a negative curve in them, different from the exceptional curve of the blowup.

We consider a parameter space of all rational triangles, and within this space, we study how the negative curves and the MDS property vary. The main theme is that almost all known results about the MDS property can be explained by the variation of negative curves in the parameter space. We attempt to classify all negative curves. In particular, we construct two new infinite families of them, one of which is the first known infinite family of special negative curves.

For a compact oriented smooth n-manifold M and a codimension-1 homology class $\mathit{\varphi }\in H_{\mathit{n}-1}(\mathit{M},\partial \mathit{M})$, we investigate a simplicial complex ${\mathcal{S}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ relating the properly embedded hypersurfaces in M representing ϕ. Its definition is akin to that of other classical complexes, such as the curve complex of a surface or the Kakimizu complex of a knot, with the difference that hypersurfaces are not taken up to isotopy.

We prove that ${\mathcal{S}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ is connected and simply connected in every dimension n. We also show the connectedness of a similar complex ${\mathcal{T}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ adapted to the three-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes $\mathcal{S}(\mathit{M},\mathit{\varphi })$ and $\mathcal{T}(\mathit{M},\mathit{\varphi })$ where hypersurfaces are taken up to isotopy, and for $\mathit{n}=2$, the simple connectedness result carries over as well. We also briefly discuss extensions to a context studied by Turaev, where regular graphs in 2-complexes are used to represent one-dimensional cohomology classes.

We finish with two applications: we give an alternative proof of the fact that all Seifert surfaces for a fixed knot in a rational homology sphere are tube-equivalent, and we use the connectedness of ${\mathcal{T}}^{\mathrm{†}}(\mathit{M},\mathit{\varphi })$ to define a new ${\ell }^2$-invariant of two-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected 3-manifolds with empty or toroidal boundary.

Let $(\mathit{C},{\mathit{p}}_1,\dots ,{\mathit{p}}_{\mathit{n}})$ be a fixed general pointed curve, and let $(\mathit{X},{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})$ be a smooth hypersurface of degree e and dimension r with n general points. We consider the problem of enumerating maps $\mathit{f}:\mathit{C}\to \mathit{X}$ of degree d (as measured in the ambient projective space) such that $\mathit{f}({\mathit{p}}_{\mathit{i}})={\mathit{x}}_{\mathit{i}}$. When e is small compared to r and d is large compared to g, e, and r, these numbers have been computed first by passing to a virtual count in Gromov–Witten theory obtained by Buch–Pandharipande and then by showing (in the work of the author with Pandharipande) that the virtual counts are enumerative via an analysis of boundary contributions in the moduli space of stable maps. In this note, we give a simpler computation via projective geometry.

We define a Rasmussen s-invariant over the coefficient ring $\mathbb{Z}$ and show how it is related to the s-invariants defined over a field. A lower bound for the slice genus of a knot arising from it is obtained, and we give examples of knots for which this lower bound is better than all lower bounds coming from the s-invariants over fields. We also compare it to the Lipshitz–Sarkar refinement related to the first Steenrod square.

A ${\mathbb{Z}}^{\mathit{d}}$-graded differential R-module is a ${\mathbb{Z}}^{\mathit{d}}$-graded R-module D with a morphism $\mathit{\delta }:\mathit{D}\to \mathit{D}$ such that ${\mathit{\delta }}^2=0$. For $\mathit{R}=\mathit{k}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{d}}]$, this paper establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology $ker\mathit{\delta }/im\mathit{\delta }$ of D is nonzero and finite dimensional over k, there is an inequality ${rk}_{\mathit{R}}\mathit{D}\ge 2^{\mathit{d}}$.

A conjecture, known as the Shokurov–Kollár connectedness principle, predicts the following. Let $(\mathit{X},\mathit{B})$ be a pair, and let $\mathit{f}:\mathit{X}\to \mathit{S}$ be a contraction with $-({\mathit{K}}_{\mathit{X}}+\mathit{B})$ nef over S; then, for any point $\mathit{s}\in \mathit{S}$, the intersection ${\mathit{f}}^{-1}(\mathit{s})\cap Nklt(\mathit{X},\mathit{B})$ has at most two connected components, where $Nklt(\mathit{X},\mathit{B})$ denotes the non-klt locus of $(\mathit{X},\mathit{B})$. This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove that this conjecture holds for threefolds in positive and mixed characteristic, where the residue fields are assumed to have characteristic $\mathit{p}>5$. Under the same assumptions, we characterize the cases in which $Nklt(\mathit{X},\mathit{B})$ fails to be connected.

The level N elliptic KZB connection is a flat connection over the universal elliptic curve in level N with its N-torsion sections removed. Its fiber over the point $(\mathit{E},\mathit{x})$ is the unipotent completion of ${\mathit{\pi }}_1(\mathit{E}-\mathit{E}[\mathit{N}],\mathit{x})$. It was constructed by Calaque and Gonzalez. In this paper, we show that the connection underlies an admissible variation of mixed Hodge structure and that it degenerates to the cyclotomic KZ connection over the singular fibers of the compactified universal elliptic curve. These are the first steps in a larger project to compute the action of the Galois group of mixed Tate motives unramified over $\mathbb{Z}[{\mathit{\mu }}_{\mathit{N}},1/\mathit{N}]$ on the unipotent fundamental group of ${\mathbb{P}}^1-\{0,{\mathit{\mu }}_{\mathit{N}},\infty \}$ and to better understand Goncharov’s higher cyclotomy.

Let k be a field of positive characteristic p with degree of imperfection 1 and a nontrivial discrete valuation. We show that the Galois and flat cohomology of unipotent k-groups over k of dimension $<\mathit{p}-1$ are finite (in fact, trivial) if and only if they are k-split. We give some examples to show that the bound $\mathit{p}-1$ is best possible and present several applications to local-global principles.

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk–Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the ham sandwich theorem and (iv) the Lyusternik–Shnirel’man–Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.

The aim of this paper is a construction of new explicit annihilators of the minus part of the ideal class group of an imaginary Abelian number field M, that is, annihilators that are outside of the Stickelberger ideal, their usual source. This construction works for quite a large class of Abelian fields M; a sufficient condition to get a new annihilator is that there is an odd prime $\ell \mid [\mathit{M}:\mathbb{Q}]$, unramified in $\mathit{M}/\mathbb{Q}$, and two primes q and ${\mathit{q}}^{\prime }$ ramifying in $\mathit{M}/\mathbb{Q}$, having their decomposition groups ${\mathit{D}}_{\mathit{q}}\subseteq {\mathit{D}}_{{\mathit{q}}^{\prime }}$ cyclic of ℓ-power order.

We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.

We improve on the result of Hacon and Witaszek by showing that the MMP for semistable fourfolds in mixed characteristic terminates in several new situations. In particular, we show the validity of the MMP for strictly semistable fourfolds over excellent Dedekind schemes globally when the residue fields are perfect and have characteristics $\mathit{p}>5$.

Given $\mathrm{\Sigma }\subset \mathit{R}:=\mathbb{K}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{k}}]$, where $\mathbb{K}$ is a field, any finite collection of linear forms, some possibly proportional, and any $1\le \mathit{a}\le |\mathrm{\Sigma }|$, we prove that ${\mathit{I}}_{\mathit{a}}(\mathrm{\Sigma })$, the ideal generated by all a-fold products of Σ, has linear graded free resolution, and we exhibit a combinatorial primary decomposition for ${\mathit{I}}_{\mathit{a}}(\mathrm{\Sigma })$. The linear graded free resolution allows us to determine a generating set for the defining ideal of the Orlik–Terao algebra of the second order of a line arrangement in ${\mathbb{P}}_{\mathbb{K}}^2$, and to conclude that for the case $\mathit{k}=3$, and Σ defining such a line arrangement, the ideal ${\mathit{I}}_{|\mathrm{\Sigma }|-2}(\mathrm{\Sigma })$ is of fiber type. Finally, with the help of the primary decomposition, we prove several conjectures of symbolic powers for the defining ideals of star configurations of any codimension c, a special class of ideals generated by a-fold products of linear forms.

We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces that recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple $(\mathit{G},\mathit{X},\mathit{H})$ consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H that is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of $\mathrm{Out}({\mathit{F}}_{\mathit{n}})$, and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich–Rafi and Dowdall–Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.

In [Xu16], Xu defines the dlt motivic zeta function associated with a regular function f on a smooth variety X over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser [DL98]. The dlt motivic zeta function is defined on a dlt modification via a Denef–Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in [Xu16], and that it is therefore not well defined.

In response to W. Kuperberg’s question, for every dimension at least two, there are uncountably many compact metric spaces, each of which (1) has the shape of the sphere of that dimension and (2) contains an infinite sequence of pairwise disjoint closed subsets irreducibly having nonzero top-dimensional integer Čech–Alexander–Spanier cohomology.

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 $\le 3$.

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 study trisections of smooth compact nonorientable 4-manifolds and introduce trisections of nonorientable 4-manifolds with boundary. In particular, we prove a nonorientable analogue of a classical theorem of Laudenbach–Poénaru and analogues for some well-known theorems from 3-manifold topology. As a consequence, trisection diagrams and Kirby diagrams of closed nonorientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of nonorientable 4-manifolds with many examples.

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.

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.

The Friedman–Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).

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.

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 study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective.

We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding.

Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a ${\mathit{C}}^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with ${\mathit{C}}^2$ boundary $\mathit{D}\subset {\mathbb{C}}^{\mathit{n}}$, we show the existence of global holomorphic coordinate systems defined on $\bar{\mathit{D}}$ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small ${\mathit{C}}^2$ perturbation of $\partial \mathit{D}$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a ${\mathit{C}}^2$ real hypersurface $\mathit{M}\subset {\mathbb{C}}^{\mathit{n}}$, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.