Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.

We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results.

If $R$ is prime radical and $\delta$ is a derivation of $R$, then the differential polynomial ring $R[X;\delta]$ is locally nilpotent. This answers an open question posed in [41].

The nil radical of a differential polynomial ring $R[X;\delta]$ takes the form $I[X;\delta]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed in [30] for algebras over infinite fields.

If $R$ is a graded algebra generated in degree $1$ over a field of characteristic zero and $\delta $ is a grading preserving derivation on $R$, then the Jacobson radical of $R$ is $\delta$-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp.

Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent.

The power series ring $R[[X;\sigma,\delta]]$ is well-defined whenever $\delta$ is a locally nilpotent $\sigma$-derivation; this answers a conjecture from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on $\delta$.

In this paper we present new examples of simple $p$-local compact groups for all odd primes. We also develop the necessary tools to show saturation, simpleness, and the non-realizability as $p$-compact groups or compact Lie groups, which can be applied in a more general framework.

In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $\beta_2$ coefficients. In the present paper we construct some counterexamples which show that a similar characterization does not hold for the $\beta_p$ coefficients with $p\neq2$. This is in strong contrast with what happens in the case of uniform $n$-rectifiability. In the second part of this paper we provide an alternative argument for a recent result of Edelen, Naber, and Valtorta about the $n$-rectifiability of measures with bounded lower $n$-dimensional density. Our alternative proof follows from a slight variant of the corona decomposition in one of the aforementioned works of Azzam and Tolsa and a suitable approximation argument.

The action of the Cremona group of rank $2$ on an infinite dimensional hyperbolic space is the main recent tool to study the Cremona group. Following~the analogy with the action of $\operatorname{PSL}(2,\mathbb{Z})$ on the Poincaré half-plane, we exhibit a fundamental domain for this action by considering a Voronoi tessellation. Then we study adjacent cells to a given cell, as well as cells that share common points in the boundary at infinity.

In this paper we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds for these quantities on the critical line (and sharpens the error terms in such estimates). Our tools come not only from number theory, but also from Fourier analysis and approximation theory. An important element in our strategy is the ability to solve a Fourier optimization problem with constraints, namely, the problem of majorizing certain real-valued even functions by bandlimited functions, optimizing the $L^1(\mathbb{R})$-error. Deriving explicit formulae for the Fourier transforms of such optimal approximations plays a crucial role in our approach.

We give a new characterization of Sobolev-Slobodeckij spaces $W^{1+s,p}$ for $n/p< 1+s$, where $n$ is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in $L^p$ and the appropriate energy is finite.

We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton [18] and recent work by the third author, Grafakos, Nakamura, and Sawano [20]. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo [10].

We prove that, if $D $ is a normal open subset of a Stein space $X$ of pure dimension such that $D$ is locally Stein at every point of $\partial D \setminus X_{\operatorname{sg}}$, then, for every holomorphic vector bundle $E$ over $D$ and every discrete subset $\Lambda $ of $D \setminus X_{\operatorname{sg}}$ whose set of accumulation points lies in $\partial D \setminus X_{\operatorname{sg}}$, there is a holomorphic section of $E$ over $D$ with prescribed values on $\Lambda$. We apply this to the local Steinness problem and domains of holomorphy.

We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle $(i\infty) - (0)$ on the Poincaré upper half plane. This analogy allows us to consider certain distributions, which can be regarded as anticyclotomic $p$-adic $L$-functions for modular forms of non-critical slope following the overconvergent strategy à la Stevens. We also give a geometric interpretation of their Gross points for the case of weight two forms. Our construction generalizes those of Bertolini-Darmon, Bertolini-Darmon-Iovita-Spiess, and Chida-Hsieh and shows a certain integrality of the interpolation formula even for non-ordinary forms.