We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(\!-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. We also prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c \gt 0$. This improves on the previously best known upper bound $O(n^2(\log\log n)^{-c})$, coming from Shkredov’s work on the corners theorem.

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime p and a positive integer n, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of p-adic integers whose entries are independent. Previously, Wood showed that as long as each entry of $X_n$ is not too concentrated on a single residue modulo p, regardless of its distribution, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen–Lenstra distribution, as $n \rightarrow \infty$. Here on the contrary, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$. When $B_n$ is a Haar-random $n \times n$ matrix over $\mathbb{Z}_p$, we explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ for every fixed n and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$. We deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$. We also show through cases the subtlety of the “universality” behavior when $B_n$ is not Haar-random.

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$, and let $\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover $p:S \to X$, where X is a hyperbolic surface. For $k \geq 2$, let $p_k: S_{k(g-1)+1} \to S_g$ be the standard k-sheeted regular cyclic cover. In this paper, we show that $\{\mathrm{LMod}_{p_k}(S_g)\}_{k \geq 2}$ forms an infinite family of self-normalising subgroups in $\mathrm{Mod}(S_g)$, which are also maximal when k is prime. Furthermore, we derive explicit finite generating sets for $\mathrm{LMod}_{p_k}(S_g)$ for $g \geq 3$ and $k \geq 2$, and $\mathrm{LMod}_{p_2}(S_2)$. For $g \geq 2$, as an application of our main result, we also derive a generating set for $\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(\iota)$, where $C_{\mathrm{Mod}(S_g)}(\iota)$ is the centraliser of the hyperelliptic involution $\iota \in \mathrm{Mod}(S_g)$. Let $\mathcal{L}$ be the infinite ladder surface, and let $q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover induced by $\langle h^{g-1} \rangle$ where h is the standard handle shift on $\mathcal{L}$. As a final application, we derive a finite generating set for $\mathrm{LMod}_{q_g}(S_g)$ for $g \geq 3$.

In an earlier work, we defined a “generalised Temperley–Lieb algebra” $TL_{r, 1, n}$ corresponding to the imprimitive reflection group G(r, 1, n) as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley–Lieb algebra $TL_{r, p, n}$ which corresponds to the complex reflection group G(r, p, n). Our definition identifies $TL_{r, p, n}$ as the fixed-point subalgebra of $TL_{r, 1, n}$ under a certain automorphism $\sigma$. We prove the cellularity of $TL_{r, p, n}$ by proving that $\sigma$ induces a special shift automorphism with respect to the cellular structure of $TL_{r, 1, n}$. We also give a description of the cell modules of $TL_{r, p, n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory.

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the logarithmic derivative $df/f$ of the polynomial. We determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by the framing of the punctured disk induced by the horizontal foliation on the translation surface.

The Mamyshev oscillator (MO) is well-known for its high modulation depth, which provides an excellent platform for achieving both high average power and short pulse durations. However, this characteristic typically limits the high-repetition-rate pulse generation. Herein, we construct an MO that achieves a gigahertz (GHz) repetition rate through harmonic mode-locking. The laser can reach up to the 93rd order, which corresponds to the repetition rate of 1.6 GHz. The maximum achieved output average power is 3 W at a repetition rate of 1.2 GHz (69th order), with the corresponding pulse duration compressed to 51 fs. To our knowledge, this is the first time that the GHz repetition rate in an MO has been obtained simultaneously with the recorded average power and pulse duration.

Based on a 4f system, a 0° reflector and a single laser diode side-pump amplifier, a new amplifier is designed to compensate the spherical aberration of the amplified laser generated by a single laser diode side-pump amplifier and enhance the power of the amplified laser. Furthermore, the role of the 4f system in the passive spherical aberration compensation and its effect on the amplified laser are discussed in detail. The results indicate that the amplification efficiency is enhanced by incorporating a 4f system in a double-pass amplifier and placing a 0° reflector only at the focal point of the single-pass amplified laser. This method also effectively uses the heat from the gain medium (neodymium-doped yttrium aluminium garnet) of the amplifier to compensate the spherical aberration of the amplified laser.

We present coherent beam combining of nanosecond pulses with 20-J energy and large beams using a Sagnac interferometer geometry based on Nd:glass rod-type amplifiers. In this study, we demonstrate that coherent beam combining is compatible with large-diameter energetic beams, presenting, therefore, an interesting and solid perspective towards the performance improvement of large-scale laser facilities, especially in terms of high-repetition-rate and high-energy operation. We demonstrate that for energy of 20 J, the coherent combination efficiency is around 92%, with high beam quality and long-term stability. A thorough temporal and spatial characterization of the system’s operation is provided to forecast the various potentialities available for large-scale facilities.

A concept for a femtosecond pulse compressor based on underdense plasma prisms is presented. An analytical model is developed to calculate the spectral phase incurred and the expected pulse compression. A 2D particle-in-cell simulation verifies the analytical model. Simulated intensities (${\sim} {10}^{16}$ W/cm2) were orders of magnitude higher than the damage threshold for conventional gratings used in chirped pulse amplification. Theoretical geometries for compact (tens of cm scale) compressors for 1, 10 and 100 PW power levels are proposed.

We propose that certain white dwarf (WD) planets, such as WD 1856+534 b, may form out of material from a stellar companion that tidally disrupts from common envelope evolution with the WD progenitor star. The disrupted companion shreds into an accretion disc, out of which a gas giant protoplanet forms due to gravitational instability. To explore this scenario, we make use of detailed stellar evolution models consistent with WD 1856+534. The minimum mass companion that produces a gravitationally unstable disc after tidal disruption is $\sim$$0.15\,\mathrm{M_\odot}$. In this scenario, WD 1856+534 b might have formed at or close to its present separation, in contrast to other proposed scenarios where it would have migrated in from a much larger separation. Planet formation from tidal disruption is a new channel for producing second-generation planets around WDs.

The propagation of multiple ultraintense femtosecond lasers in underdense plasmas is investigated theoretically and numerically. We find that the energy merging effect between two in-phase seed lasers can be improved by using two obliquely incident guiding lasers whose initial phase is $\pi$ and $\pi /2$ ahead of the seed laser. Particle-in-cell simulations show that due to the repulsion and energy transfer of the guiding laser, the peak intensity of the merged light is amplified by more than five times compared to the seed laser. The energy conversion efficiency from all incident lasers to the merged light is up to approximately 60$\%$. The results are useful for many applications, including plasma-based optical amplification, charged particle acceleration and extremely intense magnetic field generation.

Low-density polymer foams pre-ionized by a well-controlled nanosecond pulse are excellent plasma targets to trigger direct laser acceleration (DLA) of electrons by sub-picosecond relativistic laser pulses. In this work, the influence of the nanosecond pulse on the DLA process is investigated. The density profile of plasma generated after irradiating foam with a nanosecond pulse was simulated with a two-dimensional hydrodynamic code, which takes into account the high aspect ratio of interaction and the microstructure of polymer foams. The obtained plasma density profile was used as input to the three-dimensional particle-in-cell code to simulate energy, angular distributions and charge carried by the directional fraction of DLA electrons. The modelling shows good agreement with the experiment and in general a weak dependence of the electron spectra on the plasma profiles, which contain a density up-ramp and a region of near-critical electron density. This explains the high DLA stability in pre-ionized foams, which is important for applications.