In 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.

In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc \notin S$ for all $a, b, c \in S$. If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal triple-product-free set of size 1. We then derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free, and conclude with some observations and comparisons with the situation for standard product-free sets.

We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k-differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.

We consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by $\gamma_f$, in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ $g\,{:}\,(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $\gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.

This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\zeta(\sigma+it)$ for real t where $\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $\zeta(\sigma+it)$ when $\sigma>1/2$ and we show that there is a connection with the zeros of $\zeta'(\sigma+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.

Let $(X_{1},\ldots,X_{n})$ be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for $1\leq i\leq n$, $X_{i:n}$ denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is in particular proved that for $m=2,\ldots,n$, the dependence of $X_{2:m}$ on $X_{1:m}$ is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that $(X_{1:m},X_{2:m})$ is more concordant than $(X_{1},X_{2})$. It will be interesting to examine whether these results can be extended to other exchangeable models.

Erdős, Graham and Selfridge considered, for each positive integer n, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville’s problem unconditionally.

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$-extension, then it holds for almost all $\mathbb{Z}_{p}$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

We consider the problem of optimally maintaining an offshore wind farm in which major components progressively degrade over time due to normal usage and exposure to a randomly varying environment. The turbines exhibit both economic and stochastic dependence due to shared maintenance setup costs and their common environment. Our aim is to identify optimal replacement policies that minimize the expected total discounted setup, replacement, and lost power production costs over an infinite horizon. The problem is formulated using a Markov decision process (MDP) model from which we establish monotonicity of the cost function jointly in the degradation level and environment state and characterize the structure of the optimal replacement policy. For the special case of a two-turbine farm, we prove that the replacement threshold of one turbine depends not only on its own state of degradation but also on the state of degradation of the other turbine in the farm. This result yields a complete characterization of the replacement policy of both turbines by a monotone curve. The policies characterized herein can be used to optimally prescribe timely replacements of major components and suggest when it is most beneficial to share costly maintenance resources.

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upper Banach density $d^*(E) = \delta > 0$, we show, inter alia:

1. (1) if $p(x) \in K[x]$ is an intersective polynomial (i.e., p has a root modulo m for every $m \in \mathcal{O}_K$) with $p(\mathcal{O}_K) \subseteq \mathcal{O}_K$ and $r, s \in \mathcal{O}_K$ are distinct and nonzero, then for any $\varepsilon > 0$, there is a syndetic set $S \subseteq \mathcal{O}_K$ such that for any $n \in S$,

   \begin{align*}d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n)\} \subseteq E \right\} \right) > \delta^3 - \varepsilon. \end{align*}
   Moreover, if ${s}/{r} \in \mathbb{Q}$, then there are syndetically many $n \in \mathcal{O}_K$ such that
   \begin{align*}d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \right\} \right)> \delta^4 - \varepsilon; \end{align*}

2. (2) if $\{p_1, \dots, p_k\} \subseteq K[x]$ is a jointly intersective family (i.e., $p_1, \dots, p_k$ have a common root modulo m for every $m \in \mathcal{O}_K$) of linearly independent polynomials with $p_i(\mathcal{O}_K) \subseteq \mathcal{O}_K$, then there are syndetically many $n \in \mathcal{O}_K$ such that

   \begin{align*}d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \right\} \right)> \delta^{k+1} - \varepsilon. \end{align*}

These two results generalise and extend previous work of Frantzikinakis and Kra [21] and Franztikinakis [19] on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best [2] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg’s correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl’s equidistribution theorem for polynomials of several variables:

1. (3) let $d, k, l \in \mathbb{N}$. Let $(X, \mathcal{B}, \mu, T_1, \dots, T_l)$ be an ergodic, connected $\mathbb{Z}^l$-nilsystem. Let $\{p_{i,j} \;:\; 1 \le i \le k, 1 \le j \le l\} \subseteq \mathbb{Q}[x_1, \dots, x_d]$ be a family of polynomials such that $p_{i,j}\left( \mathbb{Z}^d \right) \subseteq \mathbb{Z}$ and $\{1\} \cup \{p_{i,j}\}$ is linearly independent over $\mathbb{Q}$. Then the $\mathbb{Z}^d$-sequence $\left( \prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, \dots, \prod_{j=1}^l{T_j^{p_{k,j}(n)}}x \right)_{n \in \mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure.

We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and between wheeled properads and modular operads.

We investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

This paper gives an explicit version of Selberg’s mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32\,277\log ^2 y]$ for at least half the $y\in [x,2x]$ for all $x\geq 2$, and at least one Goldbach number in $(x,x+9696 \log ^2 x]$ for all $x\geq 2$.

Let $S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $\alpha $ as $n\rightarrow \infty $ and let $\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if $A\subset [0, l]$ with $l\in A$, $\gcd A=1$ and $|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where $\lambda =\lceil {k}/{\beta }\rceil $, then $kA\cap S\neq \emptyset $ for $2<\beta \leq 3$ and $k\geq {2\beta }/{(\beta -2)}$ or for $\beta>3$ and $k\geq 3$.

In this paper, a new point process is introduced. It combines the nonhomogeneous Poisson process with the generalized Polya process (GPP) studied in recent literature. In reliability interpretation, each event (failure) from this process is minimally repaired with a given probability and GPP-repaired with the complementary probability. Characterization of the new process via the corresponding bivariate point process is presented. The mean numbers of events for marginal processes are obtained via the corresponding rates, which are used for considering an optimal replacement problem as an application.