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Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings. We study a graded version of this conjecture, and we show that it implies a novel conjecture concerning the cohomology of sheaves over non-Fano projective hypersurfaces.

We compute the Hochschild homology of the differential graded category of perfect curved modules over suitable curved rings, giving what might be termed "de Rham models" for such. This represents a generalization of previous results by Dyckerhoff, Efimov, Polishchuk, and Positselski concerning the Hochschild homology of matrix factorizations. A key ingredient in the proof is a theorem due to B. Briggs, which represents a "curved version" of a celebrated theorem of Hopkins and Neeman. The proof of Briggs' Theorem is included in an appendix to this paper., Comment: Appendix is by Benjamin Briggs

Given a complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge structure on the primitive cohomology of the associated projective hypersurface. As a consequence, we show that the Hodge conjecture for the projective hypersurface is equivalent to a dg-categorical analogue of the Hodge conjecture for the singularity category., Comment: 31 pages

Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal., Comment: 12 pages

It has been claimed that the variability of field quasars resembles gravitational lensing by a large cosmological population of free-floating planets with mass of about 10 Earths. But Galactic photometric monitoring experiments, on the other hand, exclude a large population of such planetary-mass gravitational lenses. These apparently contradictory pieces of evidence can be reconciled if the objects under consideration have a mean column-density that lies between the critical column-densities for gravitational lensing in these two contexts. Dark matter in that form is known to be weakly collisional, so that a core develops in galaxy halo density profiles, and a preferred model has already been established. Here we consider what such a model implies for Q2237+0305, which is the best-studied example of a quasar that is strongly lensed by an intervening galaxy. We construct microlensing magnification maps appropriate to the four macro-images of the quasar -- all of which are seen through the bulge of the galaxy. Each of these maps exhibits a caustic network arising from the stars, plus many small, isolated caustics arising from the free-floating "planets" in the lens galaxy. The "planets" have little influence on the magnification histograms but a large effect on the statistics of the magnification gradients. We compare our predictions to the published OGLE photometry of Q2237+0305 and find that these data are consistent with the presence of the hypothetical "planets". However, the evidence is relatively weak because the OGLE dataset is not well suited to testing our predictions and requires low-pass filtering for this application. New data from a large, space-based telescope are desirable to address this issue., Comment: v2: added DOI, updated references, published in MNRAS as accepted manuscript

Deep neural networks have been widely adopted in numerous domains due to their high performance and accessibility to developers and application-specific end-users. Fundamental to image-based applications is the development of Convolutional Neural Networks (CNNs), which possess the ability to automatically extract features from data. However, comprehending these complex models and their learned representations, which typically comprise millions of parameters and numerous layers, remains a challenge for both developers and end-users. This challenge arises due to the absence of interpretable and transparent tools to make sense of black-box models. There exists a growing body of Explainable Artificial Intelligence (XAI) literature, including a collection of methods denoted Class Activation Maps (CAMs), that seek to demystify what representations the model learns from the data, how it informs a given prediction, and why it, at times, performs poorly in certain tasks. We propose a novel XAI visualization method denoted CAManim that seeks to simultaneously broaden and focus end-user understanding of CNN predictions by animating the CAM-based network activation maps through all layers, effectively depicting from end-to-end how a model progressively arrives at the final layer activation. Herein, we demonstrate that CAManim works with any CAM-based method and various CNN architectures. Beyond qualitative model assessments, we additionally propose a novel quantitative assessment that expands upon the Remove and Debias (ROAD) metric, pairing the qualitative end-to-end network visual explanations assessment with our novel quantitative "yellow brick ROAD" assessment (ybROAD). This builds upon prior research to address the increasing demand for interpretable, robust, and transparent model assessment methodology, ultimately improving an end-user's trust in a given model's predictions.

The need for clear, trustworthy explanations of deep learning model predictions is essential for high-criticality fields, such as medicine and biometric identification. Class Activation Maps (CAMs) are an increasingly popular category of visual explanation methods for Convolutional Neural Networks (CNNs). However, the performance of individual CAMs depends largely on experimental parameters such as the selected image, target class, and model. Here, we propose MetaCAM, an ensemble-based method for combining multiple existing CAM methods based on the consensus of the top-k% most highly activated pixels across component CAMs. We perform experiments to quantifiably determine the optimal combination of 11 CAMs for a given MetaCAM experiment. A new method denoted Cumulative Residual Effect (CRE) is proposed to summarize large-scale ensemble-based experiments. We also present adaptive thresholding and demonstrate how it can be applied to individual CAMs to improve their performance, measured using pixel perturbation method Remove and Debias (ROAD). Lastly, we show that MetaCAM outperforms existing CAMs and refines the most salient regions of images used for model predictions. In a specific example, MetaCAM improved ROAD performance to 0.393 compared to 11 individual CAMs with ranges from -0.101-0.172, demonstrating the importance of combining CAMs through an ensembling method and adaptive thresholding., Comment: 9 pages

We prove that the d\'evissage property holds for periodic cyclic homology for a local complete intersection embedding into a smooth scheme. As a consequence, we show that the complexified topological Chern character maps for the bounded derived category and singularity category of a local complete intersection are isomorphisms, proving new cases of the Lattice Conjecture in noncommutative Hodge theory., Comment: 14 pages. To appear in Annals of K-theory

We present results from a radio survey for variable and transient sources on 15-min timescales, using the Australian SKA Pathfinder (ASKAP) pilot surveys. The pilot surveys consist of 505 h of observations conducted at around 1 GHz observing frequency, with a total sky coverage of 1476 deg$^2$. Each observation was tracked for approximately 8-10h, with a typical rms sensitivity of $\sim$30 $\mu$jy/beam and an angular resolution of $\sim$12 arcsec. The variability search was conducted within each 8-10h observation on a 15-min timescale. We detected 38 variable and transient sources. Seven of them are known pulsars, including an eclipsing millisecond pulsar, PSR J2039$-$5617. Another eight sources are stars, only one of which has been previously identified as a radio star. For the remaining 23 objects, 22 are associated with active galactic nuclei or galaxies (including the five intra-hour variables that have been reported previously), and their variations are caused by discrete, local plasma screens. The remaining source has no multi-wavelength counterparts and is therefore yet to be identified. This is the first large-scale radio survey for variables and transient sources on minute timescales at a sub-mJy sensitivity level. We expect to discover $\sim$1 highly variable source per day using the same technique on the full ASKAP surveys., Comment: 20 pages, 11 figures; accepted for publication in MNRAS

The Total Rank Conjecture is a coarser version of the Buchsbaum-Eisenbud-Horrocks conjecture which, loosely stated, predicts that modules with large annihilators must also have ``large'' syzygies. This conjecture was proved by the second author for rings of odd characteristic by taking advantage of the Adams operations on the category of perfect complexes with finite length homology. In this paper, we prove a stronger form of the Total Rank Conjecture for rings of characteristic two. This result may be seen as evidence that the Generalized Total Rank Conjecture (which is known to be false in odd characteristic) actually holds in characteristic two. We also formulate a meaningful extension of the Total Rank Conjecture over non Cohen-Macaulay rings and prove that this conjecture holds for any ring containing a field (independent of characteristic)., Comment: v2: minor changes in line with suggestions of referees. To appear in Duke Math. Journal. 18 pages

We prove that equivariant matrix factorization categories associated to henselian local hypersurface rings are idempotent complete, generalizing a result of Dyckerhoff in the non-equivariant case., Comment: 6 pages. To appear in the Journal of Algebra

A pulsar dynamic spectrum is an inline digital hologram of the interstellar medium; it encodes information on the propagation paths by which signals have travelled from source to telescope. To decode the hologram it is necessary to "retrieve" the phases of the wavefield from intensity measurements, which directly gauge only the field modulus, by imposing additional constraints on the model. We present a new method for phase retrieval in the context of pulsar spectroscopy. Our method makes use of the Fast Iterative Shrinkage Thresholding Algorithm (FISTA) to obtain sparse models of the wavefield in a hierarchical approach with progressively increasing depth. Once the tail of the noise distribution is reached the hierarchy terminates with a final, unregularised optimisation. The result is a fully dense model of the complex wavefield that permits the discovery of faint signals by appropriate averaging. We illustrate the performance of our method on synthetic test cases and on real data. Our algorithm, which we call H-FISTA, is implemented in the Python programming language and is freely available., Comment: Accepted for publication in MNRAS. Data available at https://www.doi.org/10.5281/zenodo.7007226 . Code available at https://github.com/sosl/H-FISTA

Fast cosmological transients such as fast radio bursts (FRBs) and gamma-ray bursts (GRBs) represent a class of sources more compact than any other cosmological object. As such they are sensitive to significant magnification via gravitational lensing from a class of lenses which are not well-constrained by observations today. Low-mass primordial black holes are one such candidate which may constitute a significant fraction of the Universe's dark matter. Current observations only constrain their density in the nearby Universe, giving fast transients from cosmological distances the potential to form complementary constraints. Motivated by this, we calculate the effect that gravitational lensing from a cosmological distribution of compact objects would have on the observed rates of FRBs and GRBs. For static lensing geometries, we rule out the prospect that all FRBs are gravitationally lensed for a range of lens masses and show that lens masses greater than $10^{-5}M_\odot$ can be constrained with 8000 un-localised high fluence FRBs at 1.4GHz, as might be detected by the next generation of FRB-finding telescopes., Comment: 10 pages (+7 Supplementary). Accepted to MNRAS

This paper extends the results of Boij, Eisenbud, Erman, Schreyer, and S\"oderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings., Comment: 26 pages. This is the final version of the article, which has now appeared in the Forum of Mathematics, Sigma

We discovered rapid intra-day variability in radio source PMN J1726+0639 at GHz frequencies, during a survey to search for such variability with the Australia Telescope Compact Array. Follow-up observations were conducted over two years and revealed a clear, repeating annual cycle in the rate, or characteristic timescale, of variability, showing that the observed variations can be attributed to scintillations from interstellar plasma inhomogeneities. The strong annual cycle includes an apparent "standstill" in April and another in September. We fit kinematic models to the data, allowing for finite anisotropy in the scintillation pattern. The cycle implies a very high degree of anisotropy, with an axial ratio of at least 13:1, and the fit is consistent with a purely one-dimensional scintillation pattern. The position angle of the anisotropy, and the transverse velocity component are tightly constrained. The parameters are inconsistent with expectations from a previously proposed model of scattering associated with plasma filaments radially oriented around hot stars. We note that evidence for a foreground interstellar cloud causing anomalous Ca II absorption towards the nearby star Rasalhague ($\alpha$ Oph) has been previously reported, and we speculate that the interstellar scintillation of PMN 1726+0639 might be associated with this nearby cloud., Comment: 7 pages, 5 figures, accepted for publication in MNRAS

Molecular hydrogen normally has only weak, quadrupole transitions between its rovibrational states, but in a static electric field it acquires a dipole moment and a set of allowed transitions. Here we use published ab initio calculations of the static electrical response tensors of the H2 molecule to construct the perturbed rovibrational eigensystem and its ground state absorptions. We restrict attention to two simple field configurations that are relevant to condensed hydrogen molecules in the interstellar medium: a uniform electric field, and the field of a point-like charge. The energy eigenstates are mixtures of vibrational and angular momentum eigenstates so there are many transitions that satisfy the dipole selection rules. We find that mixing is strongest amongst the states with high vibrational excitation, leading to hundreds of absorption lines across the optical and near infrared. These spectra are very different to that of the field-free molecule, so if they appeared in astronomical data they would be difficult to assign. Furthermore in a condensed environment the excited states likely have short lifetimes to internal conversion, giving the absorption lines a diffuse appearance. We therefore suggest electrified H2 as a possible carrier of the Diffuse Interstellar Bands (DIBs). We further argue that in principle it may be possible to account for all of the DIBs with this one carrier. However, despite electrification the transitions are not very strong and a large column of condensed H2 would be required, making it difficult to reconcile this possibility with our current understanding of the ISM., Comment: ApJ In press (33 pages, 10 figures). Full data tables available at http://www.manlyastrophysics.org

We study the influence of a cosmological population of dense gas clouds on distant sources, with emphasis on quasar optical variability. In addition to gravitational lensing such clouds affect flux measurements via refraction in the neutral gas and via dust extinction, leading to a variety of possible light curves even in the low optical depth limit. We classify and illustrate the types of light curves that can arise. For sources as large as quasars we show that gravitational lensing and extinction are the dominant effects, with gas refraction playing only a minor role. We find that clouds with mass ~10^{-4.5+/-0.5} M_\odot can reproduce the observed distribution of quasar variation amplitudes, but only if such clouds make up a large fraction of the closure density. In that case there may also be substantial extinction of distant optical sources, which can in principle be constrained by data on "standard candles" such as type Ia supernovae. Unfortunately that extinction is essentially grey, even when the material opacity is strongly wavelength dependent, making it difficult to distinguish from the influence of the background geometry. We propose a novel statistical test of the origin of quasar variability, based on the angular structure of the variation timescale for a large number of quasars distributed all over the sky. If quasar variability is primarily due to nanolensing that angular structure is expected to include a quadrupole term of amplitude $\sim5\%$, which ought to be measurable with future data from the Gaia mission., Comment: 19 pages, 6 figures plus an appendix; accepted for publication in MNRAS

'Downwards drifting' structures observed in fast radio bursts (FRBs) could naturally arise from a screen occurring after a small initial dispersion measure region. The screen imprints temporally sharp but broadband structure on the pulse that has already been dispersed, and the `structure-maximising' bulk of the dispersion measure is acquired further along the path to observer. If so, scaling of the drift rate of repeating FRBs with frequency suggests that the emission is beamed -- out of our line of sight -- with the circum-burst plasma deflecting the beam towards us. This in turn explains the observed limited, variable bandwidths of bursts despite broadband nature of the underlying emission. We summarise the geometric constraints on this simple model delivering much of the observed FRB spectro-temporal phenomenology, present generic predictions and discuss possible nature of modulation., Comment: 6 pages of text, 3 figures that capture most of the text, posted before submitting to MNRAS to gratefully gather any feedback

This work introduces a notion of complexes of maximal depth, and maximal Cohen-Macaulay complexes, over a commutative noetherian local ring. The existence of such complexes is closely tied to the Hochster's ``homological conjectures", most of which were recently settled by Andr\'e. Various constructions of maximal Cohen-Macaulay complexes are described, and their existence is applied to give new proofs of some of the homological conjectures, and also of certain results in birational geometry., Comment: 20 pages

This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good convergence properties in the rational Grothendieck group of $R$. Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety., Comment: 40 pages; 1 figure. Some parts of the text have been substantially rewritten. This work will appear in Algebra and Number Theory