We start by recalling the following theorem of Rohrlich [17]. To state it, let $\unicode[STIX]{x1D714}_{\mathfrak{z}}$ denote half of the size of the stabilizer $\unicode[STIX]{x1D6E4}_{\mathfrak{z}}$ of $\mathfrak{z}\in \mathbb{H}$ in $\text{SL}_{2}(\mathbb{Z})$ and for a meromorphic function $f:\mathbb{H}\rightarrow \mathbb{C}$ let $\text{ord}_{\mathfrak{z}}(f)$ be the order of vanishing of $f$ at $\mathfrak{z}$. Moreover, define $\unicode[STIX]{x1D6E5}(z):=q\prod _{n\geqslant 1}(1-q^{n})^{24}$, where $q:=e^{2\unicode[STIX]{x1D70B}iz}$, and set $\unicode[STIX]{x1D55B}(z):=\frac{1}{6}\log (y^{6}|\unicode[STIX]{x1D6E5}(z)|)+1$, where $z=x+iy$. Rohrlich’s theorem may be stated in terms of the Petersson inner product, denoted by $\langle ~,\,\rangle$.

The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if $(B,+)$ is nilpotent.

Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion of hole as do analytical and topological objects. However, equipping an algebraic object with a global action reveals holes in it and thanks to the homotopy theory of global actions, the holes can be described and quantified much as they are in the homotopy theory of topological spaces. Part I of this article, due to the first author, starts by recalling the notion of a global action and describes in detail the global actions attached to the general linear, elementary, and Steinberg groups. With these examples in mind, we describe the elementary homotopy theory of arbitrary global actions, construct their homotopy groups, and revisit their covering theory. We then equip the set $Um_{n}(R)$ of all unimodular row vectors of length $n$ over a ring $R$ with a global action. Its homotopy groups $\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ are christened the vector $K$-theory groups $K_{i+1}(Um_{n}(R)),i\geqslant 0$ of $Um_{n}(R)$. It is known that the homotopy groups $\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ of the general linear group $\text{GL}_{n}(R)$ viewed as a global action are the Volodin $K$-theory groups $K_{i+1,n}(R)$. The main result of Part I is an algebraic construction of the simply connected covering map $\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R)$ where $\mathit{EUm}_{n}(R)$ is the path connected component of the vector $(1,0,\ldots ,0)\in Um_{n}(R)$. The result constructs the map as a specific quotient of the simply connected covering map $St_{n}(R)\rightarrow E_{n}(R)$ of the elementary global action $E_{n}(R)$ by the Steinberg global action $St_{n}(R)$. As expected, $K_{2}(Um_{n}(R))$ is identified with $\text{Ker}(\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R))$. Part II of the paper provides an exact sequence relating stability for the Volodin $K$-theory groups $K_{1,n}(R)$ and $K_{2,n}(R)$ to vector $K$-theory groups.

Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in {\{\mathtt {0},\mathtt {1}\}}^n$ . They also stated that Holger Petersen had informed them that the constant $13$ can be reduced to $7$ . Here we show that it can be reduced to $2+\epsilon $ for any $\epsilon>0$ . The result also applies to nondeterministic automatic complexity $A_N(x)$ . In that setting the result is tight inasmuch as $A_N(x)\le n/2+1$ for all x.

We study the Hodge and weight filtrations on the localization along a hypersurface, using methods from birational geometry and the V-filtration induced by a local defining equation. These filtrations give rise to ideal sheaves called weighted Hodge ideals, which include the adjoint ideal and a multiplier ideal. We analyze their local and global properties, from which we deduce applications related to singularities of hypersurfaces of smooth varieties.

We show that relative Calabi–Yau structures on noncommutative moment maps give rise to (quasi-)bisymplectic structures, as introduced by Crawley-Boevey–Etingof–Ginzburg (in the additive case) and Van den Bergh (in the multiplicative case). We prove along the way that the fusion process (a) corresponds to the composition of Calabi–Yau cospans with ‘pair-of-pants’ ones and (b) preserves the duality between non-degenerate double quasi-Poisson structures and quasi-bisymplectic structures.

As an application, we obtain that Van den Bergh’s Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras coincide with the ones induced by the $2$-Calabi–Yau structures on (dg-versions of) these algebras.

Given a resolution of rational singularities $\pi \colon {\tilde {X}} \to X$ over a field of characteristic zero, we use a Hodge-theoretic argument to prove that the image of the functor ${\mathbf {R}}\pi _*\colon {\mathbf {D}}^{\mathrm {b}}({\tilde {X}}) \to {\mathbf {D}}^{\mathrm {b}}(X)$ between bounded derived categories of coherent sheaves generates ${\mathbf {D}}^{\mathrm {b}}(X)$ as a triangulated category. This gives a weak version of the Bondal–Orlov localization conjecture [BO02], answering a question from [PS21]. The same result is established more generally for proper (not necessarily birational) morphisms $\pi \colon {\tilde {X}} \to X$, with ${\tilde {X}}$ smooth, satisfying ${\mathbf {R}}\pi _*({\mathcal {O}}_{\tilde {X}}) = {\mathcal {O}}_X$.

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.

We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ \epsilon } < p \le 1$. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on two values for $ p = o ( (\log (n)/n)^{2/3} )$. The extent of concentration of the independence number of $ G_{n,p}$ for $ \omega (1/n) < p \le n^{-2/3}$ remains an interesting open question.

The space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the polarization of the quadratic map on metric curvature tensors that arises in the work of Hamilton on the Ricci flow. Here these algebras are studied as interesting examples of metrized commutative algebras and in low dimensions they are described concretely in terms of nonstandard commutative multiplications on self-adjoint endomorphisms. The algebra of curvature tensors on a 3-dimensional Euclidean vector space is shown isomorphic to an orthogonally invariant deformation of the standard Jordan product on $3 \times 3$ symmetric matrices. This algebra is characterized up to isomorphism in terms of purely algebraic properties of its idempotents and the spectra of their multiplication operators. On a vector space of dimension at least 4, the subspace of Weyl (Ricci-flat) curvature tensors is a subalgebra for which the multiplication endomorphisms are trace-free and the Killing type trace-form is a multiple of the nondegenerate invariant metric. This subalgebra is simple when the Euclidean vector space has dimension greater than 4. In the presence of a compatible complex structure, the analogous result is obtained for the subalgebra of Kähler Weyl curvature tensors. It is shown that the anti-self-dual Weyl tensors on a 4-dimensional vector space form a simple 5-dimensional ideal isometrically isomorphic to the trace-free part of the Jordan product on trace-free $3 \times 3$ symmetric matrices.

Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

We study a quiver description of the nested Hilbert scheme of points on the affine plane and its higher rank generalization – that is, the moduli space of flags of framed torsion-free sheaves on the projective plane. We show that stable representations of the quiver provide an ADHM-like construction for such moduli spaces. We introduce a natural torus action and use equivariant localization to compute some of their (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials. From the physics viewpoint, the quivers we study describe a class of surface defects in four-dimensional supersymmetric gauge theories in terms of nested instantons.

We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal F_s$. Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v<s$, and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal F_v$ by including a singularity in $(s-v)$. We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

Let q be a nontrivial odd prime power, and let $n \ge 2$ be a natural number with $(n,q) \ne (2,3)$ . We characterize the groups $PSL_n(q)$ and $PSU_n(q)$ by their $2$ -fusion systems. This contributes to a programme of Aschbacher aiming at a simplified proof of the classification of finite simple groups.

For any subfield $K\subseteq \mathbb{C}$, not contained in an imaginary quadratic extension of $\mathbb{Q}$, we construct conjugate varieties whose algebras of $K$-rational ($p,p$)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when $K$ is contained in an imaginary quadratic extension of $\mathbb{Q}$; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut($\mathbb{C}$)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on $H^{2}(-,\mathbb{R})$ are not (weakly) isomorphic. Using these, we detect nonhomeomorphic conjugate varieties for any fundamental group and in any birational equivalence class of dimension $\geq $10.

We compute the integral Chow ring of the moduli stack of smooth elliptic curves with n marked points for $3\leq n\leq 10$.

One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, that is, below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present complementary sharp criteria for the existence and nonexistence of zero energy ground states. Our criteria give a straightforward explanation for the folklore that there is a spectral phase transition with critical dimension four, concerning the existence versus nonexistence of zero energy ground states.

We study non-abelian versions of the Mellin transformations, originally introduced by Gabber-Loeser on complex affine tori. Our main result is a generalisation to the non-abelian context and with arbitrary coefficients of the t-exactness of Gabber-Loeser’s Mellin transformation. As an intermediate step, we obtain vanishing results for the Sabbah specialisation functors. Our main application is to construct new examples of duality spaces in the sense of Bieri-Eckmann, generalising results of Denham-Suciu.

First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem.