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The Extremely Large Telescopes (ELTs), with their large apertures and cutting-edge Multi-Conjugate Adaptive Optics (MCAO) systems, promise to deliver data that is both sharper and deeper than even the James Webb Space Telescope (JWST) across large fields. SHARP is a concept study for a near-IR (0.95-2.45 $μ$m) spectrograph specifically designed to fully exploit the collecting area and angular resolution capabilities of the upcoming ESO's ELT. The instrument concept is driven by the goal of tackling the most important questions in astrophysics and cosmology, from exploring primordial galaxies to studying the formation of young stellar object and planetary systems in the nearby dust-enshrouded regions, bridging the gap between the local and the distant Universe. This requires versatility to accommodate diverse observational needs. SHARP is composed of two main units: NEXUS, a Multi-Object Spectrograph (MOS) optimized for detecting the faintest sources, and VESPER, a multi-object Integral Field Unit (multi-IFU) designed for brighter ones. This article provides an overview of the scientific design drivers, the solutions developed to meet them, and the resulting optical design that achi

The first non-zero Laplacian eigenvalue $λ_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $κ$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $λ_1=κ$ is called Lichnerowicz sharp. In this note, we give a new proof of the classification of Lichnerowicz sharp distance-regular graphs, which was first obtained by Münch and strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, Münch, and Peyerimhoff, which required an extra spectral condition. As a key preparatory step, we provide a classification of all amply regular Terwilliger graphs with positive Lin--Lu--Yau curvature, a result that is interesting in its own right.

The persistent tension between early- and late-Universe measurements of the Hubble constant (H0) remains on of the most significant challenges in modern cosmology. The Spectroastrometry and Reverberation Mapping (SARM) method offers a promising, calibration-independent approach to address this issue by combining time-delay measurements of the Broad-Line Region (BLR) with interferometric angular size determinations. Current implementations of SARM, however, are limited by the difficulty of performing near-infrared reverberation mapping (RM) on the same emission lines observed by GRAVITY, restricting applications to only a few bright AGN. We propose using the capabilities of SHARP, the next-generation near-infrared spectrograph for the Extremely Large Telescope (ELT), to overcome these limitations. SHARP's sensitivity and multi-object spectroscopy will enable (1) efficient long-term monitoring of existing GRAVITY targets with minimal time investment, and (2) systematic RM campaigns for the fainter AGN that will be observed by GRAVITY+. These advances will give us precise infrared lags for tens of AGN, enabling geometric distance measurements and a robust, calibration-free determinati

The physical mechanisms that quench star formation in massive galaxies remain poorly understood. At cosmic noon (1<z<3), when star formation and AGN activity peak, galaxies rapidly evolve from star-forming disks into quiescent, bulge-dominated systems. While quenching correlates with stellar mass and bulge growth, the causal link between bulge assembly, star-formation suppression, and feedback processes remains unclear. Stellar population analysis from spatially resolved spectroscopy of galaxies caught during quenching is crucial to advance on this issue. We show that the SHARP-VESPER Integral-Field Spectrograph (IFS) can efficiently fill this gap by combining ELT-class sensitivity, 31 mas spatial resolution (>3x sharper than JWST) and broad near-IR wavelength coverage with 12-IFU multiplexing. This will enable, for the first time, a simultaneous bulge-disk decomposition of stellar populations and spatially resolved mapping of ionised gas in massive galaxies (log $M_*/M_{\odot}\geq 11$) at 2.2<z<3.5, targeting systems on the main sequence shortly before quenching or already in the green valley. With typical exposure times of 15 hr, we will obtain S/N>15-20 per spe

Understanding when and how massive quiescent galaxies (log(M*/Msun) > 10.5) assembled their stellar mass and quenched remains a central challenge in galaxy evolution. Spatially resolved stellar population measurements at z > 1.5 offer a uniquely powerful avenue to address this problem, as they can provide information on the radial variations in stellar age, metallicity, and enrichment histories in passive galaxies as they first emerge. In this work, we present a feasibility study quantifying the transformative capabilities of the proposed IFU SHARP/VESPER at the ELT for performing such radial mapping of stellar population gradients in passive galaxies at 1.5 < z < 3. Using the COSMOS-Web catalogue, we define a realistic population of massive quiescent systems at 1.5 < z < 3 and model representative compact and extended galaxies across this redshift range. Through detailed simulations with the official SHARP ETC, we derive the exposure times required to reach S/N = 10-15 per resolution element at key rest-frame optical wavelengths. Our results show that SHARP will routinely measure stellar population gradients out to 2Re for the majority of the population at z <

The search for and the characterization of ultra-compact dual active galactic nuclei (AGN) are among the hottest topics of current extragalactic astrophysics. These systems involve two accreting massive black holes (MBHs) embedded within the same host galaxy, with relative projected separations from a few hundred pc down to a few pc. They are central to understanding hierarchical galaxy formation, black hole growth and demographics, and accretion-feedback coupling in the most extreme interaction phases. Even more compellingly, such tight pairs are the most direct precursors of gravitationally bound binary MBHs (sub-pc scale separation), which are among the loudest emitters of gravitational waves (GWs) in the low-frequency ranges. SHARP will deliver the first statistical census and physical characterization of ultra-compact dual AGN up to cosmic distances, finally bridging the observational gap between kpc-scale pairs and sub-pc GW-emitting binaries, and enabling a breakthrough understanding of MBH growth, feedback and co-evolution across cosmic time.

Massive black hole (MBH) binaries form as a result of galaxy mergers and can coalesce into a single MBH by emitting gravitational waves detectable by LISA and pulsar timing array campaigns. Although electromagnetic observations of bound MBH binaries are extremely challenging, an indirect signature of their passage is the core scouring: a bound binary shrinks by ejecting nearby stars, creating a flat stellar density core of the size of the binary influence radius. Through this mechanism, stars on radial orbits are preferentially ejected, resulting in a central tangential anisotropy in the velocity field of stars that can be identified via IFU observations. At present, the sample of galaxies with such properties is limited by instrument resolution to the closest giant ellipticals within the nearest ~100 Mpc. The SHARP-VESPER IFU and MICADO+MORFEO instruments can work in concert to detect both these features: their unprecedented spatial resolution can allow us to detect central scourings with sizes above ~500 pc in principle up to reionization; smaller cores of ~150 pc can be detected up to z~0.14, encompassing a volume that is more than 40 times the one available at present. In addit

The stellar initial mass function (IMF), which describes the distribution of stellar masses at birth, is a fundamental ingredient in shaping galaxy evolution. Recent observations indicate that the IMF varies between galaxies, depending on their mass, morphology, and stellar content. In local early-type galaxies (ETGs), spectroscopy, dynamics, and lensing reveal bottom-heavy IMFs in dense central regions, with radial gradients toward a Milky Way-like distribution in the outskirts. Yet, the chemical enrichment of massive ETGs implies a dominant role of massive stars during their early formation phases. These findings can be reconciled if the IMF evolves over cosmic time -- initially more top-heavy to enable rapid enrichment, and later dominated by long-lived, low-mass stars. Directly measuring the IMF at z>1 is therefore essential to test such time-dependent IMF scenarios, including variations in the dwarf-to-giant and stellar mass-to-light ratios. To date, no direct observational confirmation of these IMF variations -- or of their physical origin -- has been obtained. The SHARP spectrograph on the E-ELT, with unprecedented spatial resolution and sensitivity compared to facilities

The existence of luminous quasars just a few hundred million years after the Big Bang challenges our understanding of both black hole growth and galaxy formation and evolution. These objects harbour supermassive black holes exceeding a billion solar masses (M$_{BH} > 10^{9} M_{\odot}$) by redshift $z\sim 6.5$-$7.5$, powered by extreme gas accretion. At the same time, their host galaxies are also undergoing intense star formation, consuming gas at the rate of hundreds of solar masses per year. Characterising the circumgalactic medium (CGM) and intergalactic medium (IGM) surrounding high-redshift quasars becomes an essential tool to understand the conditions that enable the rapid formation and evolution of these extreme sources. While in the last decades spatially resolved observations in the optical band have targeted CGM through Ly$α$ nebulae surrounding $z \sim 2-6$ quasars, current instrumental limitations hamper observations of high-z ($z>8$) quasars that will be discovered by Euclid/Roman/LSST surveys. Despite the large fraction of neutral hydrogen at the epoch of reionisation, in the last decade several surprising Ly$α$ detections have been obtained from sources deep in

We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere $\mathbb{S}^3$, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, Gérard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing $\mathbb{S}^3$ as the compact Lie group $\mathrm{SU}(2)$ and exploiting its representation theory. Motivated by applications to the energy-critical nonlinear Schrödinger equation (NLS) on $\mathbb{R} \times \mathbb{S}^3$, we also prove a refined anisotropic Strichartz estimate on the cylindrical space $\mathbb{R}_{x_1} \times \mathbb{T}_{x_2}$ of $L^\infty_{x_2}L^4_{t,x_1}$-type, adapted to certain spectrally localized functions. The argument relies on multiple sharp measure estimates and a robust kernel decomposition method. Combining these two key ingredients, we derive a refined bilinear Strichartz estimate on $\mathbb{R} \times \mathbb{S}^3$, which in turn yields small-data global well-posedness for the above mentioned NLS in the energy space.

The systole of a contact form $α$ is defined as the shortest period of closed Reeb orbits of $α$. Given a non-trivial $\mathbb S^1$-principal bundle over $\mathbb S^2$ with total space $M$, we prove a sharp systolic inequality for the class of tight contact form on $M$ invariant under the $\mathbb S^1$-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on $\mathbb S^2$, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.

Advancements in text-to-image diffusion models have led to significant progress in fast 3D content creation. One common approach is to generate a set of multi-view images of an object, and then reconstruct it into a 3D model. However, this approach bypasses the use of a native 3D representation of the object and is hence prone to geometric artifacts and limited in controllability and manipulation capabilities. An alternative approach involves native 3D generative models that directly produce 3D representations. These models, however, are typically limited in their resolution, resulting in lower quality 3D objects. In this work, we bridge the quality gap between methods that directly generate 3D representations and ones that reconstruct 3D objects from multi-view images. We introduce a multi-view to multi-view diffusion model called Sharp-It, which takes a 3D consistent set of multi-view images rendered from a low-quality object and enriches its geometric details and texture. The diffusion model operates on the multi-view set in parallel, in the sense that it shares features across the generated views. A high-quality 3D model can then be reconstructed from the enriched multi-view se

We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be computed by Boolean circuits of bounded depth and polynomial size. We also prove a partial converse: if a monotone graph invariant Boolean function does not have a sharp threshold then it can be computed on average by a Boolean circuit of bounded depth and polynomial size. Our general result also implies new average-case bounded depth circuit lower bounds in a variety of settings. (a) ($k$-cliques) For $k=Θ(n)$, we prove that any circuit of depth $d$ deciding the presence of a size $k$ clique in a random graph requires exponential-in-$n^{Θ(1/d)}$ size. (b)(random 2-SAT) We prove that any circuit of depth $d$ deciding the satisfiability of a random 2-SAT formula requires exponential-in-$n^{Θ(1/d)}$ size. To the best of our knowledge, this is the first bounded depth circuit lower bound for random $k$-SAT for any value of $k \geq 2.$ Our results also provide the first rigorous lower bound in agreement with a conjectured, but debated, "computational hardnes

We study the contribution of large scalar perturbations sourced by a sharp feature during cosmic inflation to the stochastic gravitational wave background (SGWB), extending our previous work to include the SGWB sourced during the inflationary era. We focus in particular on three-field inflation, since the third dynamical field is the first not privileged by the perturbations' equations of motion and allows a more direct generalization to $N$-field inflation. For the first time, we study the three-field isocurvature perturbations sourced during the feature and include the effects of isocurvature masses. In addition to a two-field limit, we find that the third field's dynamics during the feature can source large isocurvature transients which then later decay, leaving an inflationary-era-sourced SGWB as their only observable signature. We find that the inflationary-era signal shape near the peak is largely independent of the number of dynamical fields and has a greatly enhanced amplitude sourced by the large isocurvature transient, suppressing the radiation-era contribution and opening a new window of detectable parameter space with small adiabatic enhancement. The largest enhancement

In this paper, we mainly consider nonnegative weak solution to the $D^{1,p}(\R^{N})$-critical quasi-linear static Schrödinger-Hartree equation with $p$-Laplacian $-Δ_{p}$ and nonlocal nonlinearity: \begin{align*} -Δ_p u =\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \,\, \mathbb{R}^N, \end{align*} where $1<p<\frac{N}{2}$, $N\geq3$ and $u\in D^{1,p}(\R^N)$. Being different to the $D^{1,p}(\R^{N})$-critical local nonlinear term $u^{p^{\star}-1}$ with $p^{\star}:=\frac{Np}{N-p}$ investigated in \cite{CFR,LDSMLMSB,GV,Ou,BS16,VJ16} etc., since the nonlocal convolution $|x|^{-2p}*u^p$ appears in the Hartree type nonlinearity, it is impossible for us to use the scaling arguments and the Doubling Lemma as in \cite{VJ16} to get preliminary estimates on upper bounds of asymptotic behaviors for any positive solutions $u$. Moreover, it is also quite difficult to obtain the boundedness of the quasi-norm $\|u \|_{L^{s,\infty}(\R^N)}$ and hence derive the sharp estimates on upper bounds of asymptotic behaviors from the preliminary estimates as in \cite{VJ16}. Fortunately, by showing a better preliminary estimates on upper bounds of asymptotic behaviors through the De Gi

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces, symmetrisation, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ spaces. The proof of this result uses Hamilton-Jacobi inequality and Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also obtained in ${\sf RCD}(0,N)$ spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (J. Geom. Anal., 2004) on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm $\left(\int_ΩF^{N}( abla u)\;\mathrm{d}x\right)^{\frac{1}{N}}$ in the Sobolev-type space $D^{N,q}(\mathbb{R}^{N})$, $q\geq 1$, here $F:\mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of class $C^{2}(\mathbb{R}^{N}\setminus\{0\})$, which is even and positively homogeneous of degree 1, its polar $F^{0}$ represents a Finsler metric on $\mathbb{R}^{N}$. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger-Moser inequalities and discuss their sharpness under several different situations, including the case $\|F( abla u)\|_{N}\leq 1$, the case $\|F( abla u)\|_{N}^{a}+\|u\|_{q}^{b}\leq 1$, and whether they are associated with exact growth.

We establish a sharp Sobolev trace inequality on the Siegel domain $Ω_{n+1}$ involving the weighted norm-$W^{2,2}(Ω_{n+1}, ρ^{1-2[γ]})$. The inequality is closely related the realization of fractional powers of the sub-Laplacian on the Heisenberg group $H^n=\partial Ω_{n+1}$ as generalized Dirichlet-to-Neumann operators associated to the weighted poly-sublaplacian, generalizing observations of Frank--González--Monticelli--Tan.