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This paper presents the design and technical progress of a precision X-Y stage for detector dithering and flexure compensation. The stage is being developed for use in the Magellan InfraRed Multi-Object Spectrograph, MIRMOS. MIRMOS is a very large Nasmyth mounted spectrograph containing a combination of refractive, reflective and diffractive optics mounted on a long cryogenic optical bench. The instrument utilizes five science cameras, each having a custom x-y stage to control the in-plane detector position within each camera, providing both dithering capability for improved sampling, and flexure compensation to correct for image motion that results from the gravity variant operation of the instrument. Designed to operate at 120~K, the stage will accurately control detector position in two orthogonal degrees of freedom, and have manual fine adjustment features to set detector tip, tilt and piston. The piezo-driven flexure stage provides high-resolution backlash-free motion of the detector and is very compact along the optical path, keeping camera length to a minimum. A magnetoresistive bridge provides position feedback in each degree of freedom, greatly reducing hysteresis, which is common in piezoelectric actuators. The system is designed to operate in open loop using a lookup table keyed to the Nasmyth rotator angle for flexure control. Here, the optomechanical design of the stage, electrical control system, and current performance results from early prototype efforts are presented and discussed., Comment: Proceedings of SPIE Astronomical Telescopes + Instrumentation 2024, 11 pages, 10 figures

The Magellan InfraRed Multi-Object Spectrograph (MIRMOS) is a planned next generation multi-object and integral field spectrograph for the 6.5m Magellan telescopes at Las Campanas Observatory in Chile. MIRMOS will perform R$\sim$3700 spectroscopy over a simultaneous wavelength range of 0.886 - 2.404$\mu$m (Y,J,H,K bands) in addition to imaging over the range of 0.7 - 0.886$\mu$m. The integral field mode of operation for MIRMOS will be achieved via an image slicer style integral field unit (IFU) located on a linear stage to facilitate movement into the beam during use or storage while operating in multi-object mode. The IFU will provide a $\rm \sim20"\times26"$ field of view (FoV) made up of $\rm0.84"\times26"$ slices. This will be the largest FoV IFS operating at these wavelengths from either the ground or space, making MIRMOS an ideal instrument for a wide range of science cases including studying the high redshift circumgalactic medium and emission line tracers from ionized and molecular gas in nearby galaxies. In order to achieve the desired image quality and FoV while matching the focal ratio to the multi-object mode, our slicer design makes use of novel freeform surfaces for the pupil mirrors, which require the use of high precision multi-axis diamond milling to manufacture. We present here the optical design and predicted performance of the MIRMOS IFU along with a conceptual design for the opto-mechanical system., Comment: Proceedings of SPIE Astronomical Telescopes + Instrumentation 2024, 9 pages, 6 figures

If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian., Comment: 8 pages

We present the stellar metallicities and multi-element abundances (C, Mg, Si, Ca, Ti, Cr, and Fe) of 15 massive (log M/M$_\odot$=10.2-11.2) quiescent galaxies at z=1-3, derived from ultradeep JWST-SUSPENSE spectra. Compared to quiescent galaxies at z~0, these galaxies exhibit a deficiency of 0.25 dex in [C/H], 0.16 dex in [Fe/H], and 0.07 dex in [Mg/H], implying rapid formation and quenching before significant enrichment from asymptotic giant branch stars and Type Ia supernovae. Additionally, we find that galaxies that form at higher redshift have higher [Mg/Fe] and lower [Fe/H] and [Mg/H], irrespective of their observed redshift. The evolution in [Fe/H] and [C/H] is therefore primarily explained by lower redshift samples naturally including galaxies with longer star-formation timescales. On the other hand, the lower [Mg/H] can be explained by galaxies forming at earlier epochs expelling larger gas reservoirs during their quenching phase. Consequently, the mass-metallicity relation, primarily reflecting [Mg/H], is also lower at z=1-3 compared to the lower redshift relation, though the slopes are similar. Finally, we compare our results to standard stellar population modeling approaches employing solar abundance patterns and non-parametric star-formation histories (using Prospector). Our SSP-equivalent ages agree with the mass-weighted ages from Prospector, while the metallicities disagree significantly. Nonetheless, the metallicities better reflect [Fe/H] than total [Z/H]. We also find that star-formation timescales inferred from elemental abundances are significantly shorter than those from Prospector, and we discuss the resulting implications for the early formation of massive galaxies., Comment: Submitted to ApJ; 18 pages, 6 figures, 1 table

Recent stellar chemical abundance measurements of a handful of $z\sim2$ quiescent galaxies have suggested these galaxies exhibit a remarkably strong $\alpha$-enhancement compared to their local and intermediate redshift counterparts. This apparent chemical evolution following quenching suggests that even the innermost regions of massive early-type galaxies may have experienced substantial mixing of stars in mergers, challenging a purely inside-out growth model. However, larger samples are needed to determine whether a high $\alpha$-enhancement ([Mg/Fe] $\approx 0.5$) is common in $z \sim 2$ quiescent galaxies, and a comparative analysis is needed to determine whether it is consistently inferred using different stellar population synthesis models. We report age and stellar chemical abundance measurements for a sample of four gravitationally lensed quiescent galaxies at $z\sim2.1-2.65$ based on Magellan/FIRE spectroscopy. For three of these galaxies we constrain the $\alpha$-enhancement, and in two cases we measure high values comparable to earlier results when the spectra are analyzed consistently. We also find that the choice of modeling approach can exert a significant effect on the measured abundances. This model dependence can be partly, but not entirely, explained by the complex abundance patterns of $\alpha$ elements in galaxies, which has been observed at lower redshifts and in one $z \sim 2$ quiescent galaxy. Our investigation highlights the importance of independently varying abundance of $\alpha$ elements when fitting the spectra of such galaxies. Observations with JWST will soon deliver precise and spatially resolved abundances of these and other quiescent galaxies at cosmic noon, opening a new window into their evolution., Comment: 10 pages, submitted to The Astrophysical Journal

We present an overview and first results from the Spectroscopic Ultradeep Survey Probing Extragalactic Near-infrared Stellar Emission (SUSPENSE), executed with NIRSpec on JWST. The primary goal of the SUSPENSE program is to characterize the stellar, chemical, and kinematic properties of massive quiescent galaxies at cosmic noon. In a single deep NIRSpec/MSA configuration, we target 20 distant quiescent galaxy candidates ($z=1-3$, $H_{AB}\le23$), as well as 53 star-forming galaxies at $z=1-4$. With 16~hr of integration and the G140M-F100LP dispersion-filter combination, we observe numerous Balmer and metal absorption lines for all quiescent candidates. We derive stellar masses (log$M_*/M_{\odot}\sim10.2-11.5$) and detailed star-formation histories (SFHs) and show that all 20 candidate quiescent galaxies indeed have quenched stellar populations. These galaxies show a variety of mass-weighted ages ($0.8-3.3$~Gyr) and star formation timescales ($\sim0.5-4$~Gyr), and four out of 20 galaxies were already quiescent by $z=3$. On average, the $z>1.75$ $[z<1.75]$ galaxies formed 50\% of their stellar mass before $z=4$ $[z=3]$. Furthermore, the typical SFHs of galaxies in these two redshift bins ($z_{\text{mean}}=2.2~[1.3]$) indicate that galaxies at higher redshift formed earlier and over shorter star-formation timescales compared to lower redshifts. Although this evolution is naturally explained by the growth of the quiescent galaxy population over cosmic time, number density calculations imply that mergers and/or late-time star formation also contribute to the evolution. In future work, we will further unravel the early formation, quenching, and late-time evolution of these galaxies by extending this work with studies on their chemical abundances, resolved stellar populations and kinematics., Comment: Accepted in ApJ; 25 pages, 14 figures, 2 tables (excluding appendices)

We consider the hypergraph Tur\'an problem of determining $\mathrm{ex}(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then $\mathrm{ex}(n, S^d) \geq \Omega(n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$. Furthermore, this lower bound holds unconditionally for 2-LC spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on $\mathrm{ex}(n, S^d)$ of $O(n^{d + 1 - 1/2^{d - 1}})$ using a simple induction argument. We conjecture that the upper bound can be improved to match the conditional lower bound., Comment: 9 pages

A recent conjecture of Chudnovsky and Nevo asserts that flag triangulations of spheres always have linear-sized independent sets, with a precisely conjectured proportion depending on the dimension. For dimensions one and two, the lower bound of their conjecture basically follow from constant bounds on the chromatic number of flag triangulations of $S^1$ and $S^2$. This raises a natural question that does not appear to have been considered: For each $d$ is there a constant upper bound for the chromatic number of flag triangulations of $S^d$? Here we show that the answer to this question is no, and use results from Ramsey theory to construct flag triangulations of 3-spheres on $n$ vertices with chromatic number at least $\widetilde{\Omega}(n^{1/4})$., Comment: 8 pages, 2 figures

The connection between galaxies and dark matter halos is often quantified using the stellar mass-halo mass (SMHM) relation. Optical and near-infrared imaging surveys have led to a broadly consistent picture of the evolving SMHM relation based on measurements of galaxy abundances and angular correlation functions. Spectroscopic surveys at $z \gtrsim 2$ can also constrain the SMHM relation via the galaxy autocorrelation function and through the cross-correlation between galaxies and Ly$\alpha$ absorption measured in transverse sightlines; however, such studies are very few and have produced some unexpected or inconclusive results. We use $\sim$3000 spectra of $z\sim2.5$ galaxies from the Lyman-alpha Tomography IMACS Survey (LATIS) to measure the galaxy-galaxy and galaxy-Ly$\alpha$ correlation functions in four bins of stellar mass spanning $10^{9.2} \lesssim M_* / M_{\odot} \lesssim 10^{10.5}$. Parallel analyses of the MultiDark N-body and ASTRID hydrodynamic cosmological simulations allow us to model the correlation functions, estimate covariance matrices, and infer halo masses. We find that results of the two methods are mutually consistent and are broadly in accord with standard SMHM relations. This consistency demonstrates that we are able to accurately measure and model Ly$\alpha$ transmission fluctuations $\delta_F$ in LATIS. We also show that the galaxy-Ly$\alpha$ cross-correlation, a free byproduct of optical spectroscopic galaxy surveys at these redshifts, can constrain halo masses with similar precision to galaxy-galaxy clustering., Comment: Accepted to ApJ

We present the stellar mass - stellar metallicity relation for 3491 star-forming galaxies at $2 \lesssim z \lesssim 3$ using rest-frame far-ultraviolet (FUV) spectra from the Ly$\alpha$ Tomography IMACS Survey (LATIS). We fit stellar population synthesis models from the Binary Population And Spectral Synthesis code (BPASS v$2.2.1$) to medium resolution (R $\sim 1000$) and high signal-to-noise ($>30$ per 100 km/s over a wavelength range of 1221 - 1800 \r{A}) composite spectra of galaxies in bins of stellar mass to determine their stellar metallicity, primarily tracing $\rm Fe/H$. We find a strong correlation between stellar mass and stellar metallicity, with stellar metallicity monotonically increasing with stellar mass at low masses and flattening at high masses ($M_* \gtrsim 10^{10.3} M_\odot$). Additionally, we compare our stellar metallicity measurements with the gas-phase oxygen abundance of galaxies at similar redshift and estimate the average $\rm [\alpha/Fe] \sim 0.6$. Such high $\alpha$-enhancement indicates that high-redshift galaxies have not yet undergone significant iron enrichment through Type Ia supernovae. Moreover, we utilize an analytic chemical evolution model to constrain the mass loading parameter of galactic winds as a function of stellar mass. We find that as the stellar mass increases, the mass loading parameter decreases. The parameter then flattens or reaches a turning point at around $M_* \sim 10^{10.5} M_\odot$. Our findings may signal the onset of black hole-driven outflows at $z \sim 2.5$ for galaxies with $M_* \gtrsim 10^{10.5} M_\odot$., Comment: 21 pages, 17 figures, 3 tables, accepted for publication in ApJ

We forecast the prospects for cross-correlating future line intensity mapping (LIM) surveys with the current and future Ly-$\alpha$ forest data. We use large cosmological hydrodynamic simulations to model the expected emission signal for the CO rotational transition in the COMAP LIM experiment at the 5-year benchmark and the Ly-$\alpha$ forest absorption signal for various surveys, including eBOSS, DESI, and PFS. We show that CO$\times$Ly-$\alpha$ forest can significantly enhance the detection signal-to-noise ratio of CO, with a $200$ to $300 \%$ improvement when cross-correlated with the forest observed in the Prime Focus Spectrograph (PFS) survey and a $50$ to $75\%$ enhancement for the currently available eBOSS or the upcoming DESI observations. We compare to the signal-to-noise improvements expected for a galaxy survey and show that CO$\times$Ly-$\alpha$ is competitive with even a spectroscopic galaxy survey in raw signal-to-noise. Furthermore, our study suggests that the clustering of CO emission is tightly constrained by CO$\times$Ly-$\alpha$ forest, due to the increased signal-to-noise ratio and the simplicity of Ly-$\alpha$ absorption power spectrum modeling. Any foreground contamination or systematics are expected not to be shared between LIM surveys and Ly-$\alpha$ forest observations; this provides an unbiased inference. Our findings highlight the potential benefits of utilizing the Ly-$\alpha$ forest to aid in the initial detection of signals in line intensity experiments. For example, we also estimate that [CII]$\times$Ly-$\alpha$ forest measurements from EXCLAIM and DESI/eBOSS, respectively, should have a larger signal-to-noise ratio than planned [CII]$\times$quasar observations by about an order of magnitude. Our results can be readily applied to actual data thanks to the observed quasar spectra in eBOSS Stripe 82, which overlaps with several LIM surveys., Comment: Codes and the produced data are available at https://github.com/qezlou/lali

We study the intersecting family process initially studied in \cite{BCFMR}. Here $k=k(n)$ and $E_1,E_2,\ldots,E_m$ is a random sequence of $k$-sets from $\binom{[n]}{k}$ where $E_{r+1}$ is uniformly chosen from those $k$-sets that are not already chosen and that meet $E_i,i=1,2,\ldots,r$. We prove some new results for the case where $k=cn^{1/3}$ and for the case where $k\gg n^{1/2}$.

For $X \sim X(n; 1, n^{-\alpha_1}, n^{-\alpha_2}, ...)$ in the multiparameter random simplicial complex model we establish necessary and sufficient strict inequalities on the $\alpha_i$'s to linearly embed the complex into $\mathbb{R}^{2d}$., Comment: 15 pages. There was an error for the odd-dimensional case in the previous version. This version proves the threshold for embeddability into even dimensions only. Section 2 has also been expanded to include more detail

We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random Borsuk--Ulam theorem. We apply this to derive a structural result for pairs of non-adjacent cliques in Erd\H{o}s--R\'{e}nyi random graphs., Comment: 25 pages

We present new observations of MRG-M2129, a quiescent galaxy at z = 2.15 with the Atacama Large Millimeter/submillimeter Array (ALMA). With the combination of the gravitational lensing effect by the foreground cluster and the angular resolution provided by ALMA, our data reveal 1.2 mm continuum emission at $\sim130$ pc angular resolution. Compact dust continuum is detected at 7.9 $\sigma$ in the target but displaced from its stellar peak position by $62 \pm 38$ mas, or $\sim169 \pm 105$ pc in the source plane. We find considerably high dust-to-stellar mass ratio, $4 \times 10^{-4}$. From non-detection of the [C i] 3P2 -> 3P1 line, we then derive $3 \sigma$ upper limits on the molecular gas-to-dust mass ratio $\delta_\mathrm{GDR} < 60$ and the molecular gas-to-stellar mass ratio fH2 < 2.3%. The derived $\delta_\mathrm{GDR}$ is $>2\times$ smaller than the typical value assumed for quiescent galaxies in the literature. Our study supports that there exists a broad range of $\delta_\mathrm{GDR}$ and urges submillimeter follow-up observations of quenching/recently quenched galaxies at similar redshifts. Based on the derived low $\delta_\mathrm{GDR}$ and other observed dust properties, we argue that the central black hole is still active and regulates star formation in the system. Our study exhibits a rare case of a gravitationally lensed type 2 QSO harbored by a quiescent galaxy., Comment: Accepted for publication in ApJ

Galaxy protoclusters, which will eventually grow into the massive clusters we see in the local universe, are usually traced by locating overdensities of galaxies. Large spectroscopic surveys of distant galaxies now exist, but their sensitivity depends mainly on a galaxy's star formation activity and dust content rather than its mass. Tracers of massive protoclusters that do not rely on their galaxy constituents are therefore needed. Here we report observations of Lyman-$\alpha$ absorption in the spectra of a dense grid of background galaxies, which we use to locate a substantial number of candidate protoclusters at redshifts 2.2-2.8 via their intergalactic gas. We find that the structures producing the most absorption, most of which were previously unknown, contain surprisingly few galaxies compared to the dark matter content of their analogs in cosmological simulations. Nearly all are expected to be protoclusters, and we infer that half of their expected galaxy members are missing from our survey because they are unusually dim at rest-frame ultraviolet wavelengths. We attribute this to an unexpectedly strong and early influence of the protocluster environment on the evolution of these galaxies that reduced their star formation or increased their dust content., Comment: Published in Nature. The version of record is available at https://www.nature.com/articles/s41586-022-04681-6 or view-only at https://rdcu.be/cPIIX

The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} - (\log n)^{-\epsilon}) n^d$. Up to the first order term, this is the best possible lower bound for the maximum diameter of a $d$-complex on $n$ vertices as a simple volume argument shows that the diameter of a $d$-dimensional simplicial complex is at most $ \frac{1}{d} \binom{n}{d}$. We also find the right first-order asymptotics for the maximum diameter of a $d$-pseudomanifold on $n$ vertices., Comment: 19 pages

Rotation curves of galaxies probe their total mass distributions, including dark matter. Dwarf galaxies are excellent systems to investigate the dark matter density distribution, as they tend to have larger fractions of dark matter compared to higher mass systems. The core-cusp problem describes the discrepancy found in the slope of the dark matter density profile in the centres of galaxies ($\beta^*$) between observations of dwarf galaxies (shallower cores) and dark matter-only simulations (steeper cusps). We investigate $\beta^*$ in six nearby spiral dwarf galaxies for which high-resolution CO $J=1-0$ data were obtained with ALMA. We derive rotation curves and decompose the mass profile of the dark matter using our CO rotation curves as a tracer of the total potential and 4.5$\mu$m photometry to define the stellar mass distribution. We find $\langle\beta^*\rangle = 0.6$ with a standard deviation of $\pm0.1$ among the galaxies in this sample, in agreement with previous measurements in this mass range. The galaxies studied are on the high stellar mass end of dwarf galaxies and have cuspier profiles than lower mass dwarfs, in agreement with other observations. When the same definition of the slope is used, we observe steeper slopes than predicted by the FIRE and NIHAO simulations. This may signal that these relatively massive dwarfs underwent stronger gas inflows toward their centres than predicted by these simulations, that these simulations over-predict the frequency of accretion or feedback events, or that a combination of these or other effects are at work., Comment: Accepted for publication in MNRAS, 21 pages (4 pages of appendix material), 11 figures (3 appendix figures), code and data products available here: https://github.com/rclevy/RotationCurveTiltedRings

Ly-$\alpha$ tomography surveys have begun to produce three-dimensional (3D) maps of the intergalactic medium (IGM) opacity at $z \sim 2.5$ with Mpc resolution. These surveys provide an exciting new way to discover and characterize high-redshift overdensities, including the progenitors of today's massive groups and clusters of galaxies, known as protogroups and protoclusters. We use the IllustrisTNG-$300$ hydrodynamical simulation to build mock maps that realistically mimic those observed in the Ly-$\alpha$ Tomographic IMACS Survey (LATIS). We introduce a novel method for delineating the boundaries of structures detected in 3D Ly-$\alpha$ flux maps by applying the watershed algorithm. We provide estimators for the dark matter masses of these structures (at $z\sim2.5$), their descendant halo masses at $z=0$, and the corresponding uncertainties. We also investigate the completeness of this method for the detection of protogroups and protoclusters. Compared to earlier work, we apply and characterize our method over a wider mass range that extends to massive protogroups. We also assess the widely used fluctuating Gunn-Peterson approximation (FGPA) applied to dark-matter-only simulations; we conclude while it is adequate for estimating the Ly-$\alpha$ absorption signal from moderate-to-massive protoclusters ($ \gtrsim 10^{14.2} M_{\odot}$), it artificially merges a minority of lower-mass structures with more massive neighbors. Our methods will be applied to current and future Ly-$\alpha$~tomography surveys to create catalogs of overdensities and study environment-dependent galaxy evolution in the Cosmic~Noon~era., Comment: 18 pages, 18 figures, codes available at https://github.com/mahdiqezlou/LyTomo-Watershed

For a $k$-uniform hypergraph $\mathcal{H}$ on vertex set $\{1, ..., n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erd\H{o}s--R\'{e}nyi random $k$-uniform hypergraph, $\text{coker}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = \omega(1/n^{k - 1})$, $\text{coker}(M(\mathcal{H}))$ is torsion-free., Comment: 10 pages

The stellar initial mass function (IMF) is a fundamental property in the measurement of stellar masses and galaxy star formation histories. In this work we focus on the most massive galaxies in the nearby universe $\log(M_{\star}/M_{\odot})>11.2$. We obtain high quality Magellan/LDSS-3 long slit spectroscopy with a wide wavelength coverage of $0.4\mu{\rm m}-1.01\mu{\rm m}$ for 41 early-type galaxies (ETGs) in the MASSIVE survey, and derive high S/N spectra within an aperture of $R_{\rm e}/8$. Using detailed stellar synthesis models, we constrain the elemental abundances and stellar IMF of each galaxy through full spectral modeling. All the ETGs in our sample have an IMF that is steeper than a Milky Way (Kroupa) IMF. The best-fit IMF mismatch parameter, $\alpha_{\rm IMF}=(M/L)/(M/L)_{\rm MW}$, ranges from 1.12 to 3.05, with an average of $\langle \alpha_{\rm IMF} \rangle=1.84$, suggesting that on average, the IMF is more bottom-heavy than Salpeter. Comparing the estimated stellar mass with the dynamical mass, we find that most galaxies have stellar masses smaller than their dynamical masses within the $1\sigma$ uncertainty. We complement our sample with lower-mass galaxies from the literature, and confirm that $\log(\alpha_{\rm IMF})$ is positively correlated with $\log(\sigma)$, $\log(M_{\star})$, and $\log(M_{\rm dyn})$. The IMF in the centers of more massive ETGs is more bottom-heavy. In addition, we find that $\log(\alpha_{\rm IMF})$ is positively correlated with both [Mg/Fe] and the estimated total metallicity [Z/H]. We find suggestive evidence that the effective stellar surface density $\Sigma_{\rm Kroupa}$ might be responsible for the variation of $\alpha_{\rm IMF}$. We conclude that $\sigma$, [Mg/Fe] and [Z/H] are the primary drivers of the global stellar IMF variation., Comment: 23 pages, 11 figures, submitted to ApJ