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This is a written version of the invited lecture at the 9th European Congress of Mathematics in July 2024 in Sevilla. We review certain new symmetries of Grothendieck rings that have emerged in representation theory., Comment: 16 pages, to appear in Proceedings of the 9th European Congress of Mathematics

Purpose: Apparent diffusion coefficient (ADC) maps derived from diffusion weighted (DWI) MRI provides functional measurements about the water molecules in tissues. However, DWI is time consuming and very susceptible to image artifacts, leading to inaccurate ADC measurements. This study aims to develop a deep learning framework to synthesize ADC maps from multi-parametric MR images. Methods: We proposed the multiparametric residual vision transformer model (MPR-ViT) that leverages the long-range context of ViT layers along with the precision of convolutional operators. Residual blocks throughout the network significantly increasing the representational power of the model. The MPR-ViT model was applied to T1w and T2- fluid attenuated inversion recovery images of 501 glioma cases from a publicly available dataset including preprocessed ADC maps. Selected patients were divided into training (N=400), validation (N=50) and test (N=51) sets, respectively. Using the preprocessed ADC maps as ground truth, model performance was evaluated and compared against the Vision Convolutional Transformer (VCT) and residual vision transformer (ResViT) models. Results: The results are as follows using T1w + T2-FLAIR MRI as inputs: MPR-ViT - PSNR: 31.0 +/- 2.1, MSE: 0.009 +/- 0.0005, SSIM: 0.950 +/- 0.015. In addition, ablation studies showed the relative impact on performance of each input sequence. Both qualitative and quantitative results indicate that the proposed MR- ViT model performs favorably against the ground truth data. Conclusion: We show that high-quality ADC maps can be synthesized from structural MRI using a MPR- VCT model. Our predicted images show better conformality to the ground truth volume than ResViT and VCT predictions. These high-quality synthetic ADC maps would be particularly useful for disease diagnosis and intervention, especially when ADC maps have artifacts or are unavailable., Comment: arXiv admin note: text overlap with arXiv:2311.15044

Herschel operated as an observatory, therefore it did not cover the whole sky, but still observed ~8% of it. The first version of an overall Herschel/PACS Point Source Catalogue was released in 2017. The data are still unique and are very important for research, especially because no new far-infrared mission is foreseen for at least the next decade. In the framework of the NEMESIS project, we revisited all the photometric observations obtained by the PACS instrument on-board of the Herschel space observatory. We aimed to build the most complete and most accurate Herschel/PACS catalogue to date. Our primary goal was to increase the number of real sources, and decrease the number of spurious sources identified on a strongly variable background. Our goal was to build a blind catalogue, meaning that source extraction is conducted without relying on prior detections at various wavelengths, allowing us to detect sources never catalogued before. We define a hybrid strategy that includes classical and ML source identification and characterisation methods, providing catalogues at much higher completeness levels than before. Quality assessment also involves ML techniques. Our source extraction methodology facilitates a systematic and impartial comparison of sensitivity levels across various Herschel fields, a task that was typically beyond the scope of individual programs. We created a high-reliability and a rejected source catalogue for each PACS passband, i.e., at 70, 100, and 160 {\mu}m. With the high-reliability catalogue, we managed to significantly increase the completeness in all bands. At the same time, while the number of high-reliability detections decreased, the number of sources matching with existing catalogues increased, suggesting that the purity is also higher than before. The photometric accuracy of our pipeline is ~1% based on comparison with the standard star models., Comment: accepted for publication in Astronomy&Astrophysics, article reference: aa50032-24

We present TRACE, an almost time-reversible hybrid integrator for the planetary N-body problem. Like hybrid symplectic integrators, TRACE can resolve close encounters between particles while retaining many of the accuracy and speed advantages of a fixed time-step symplectic method such the Wisdom-Holman map. TRACE switches methods time-reversibly during close encounters following the prescription of Hernandez & Dehnen. In this paper we describe the derivation and implementation of TRACE and study its performance for a variety of astrophysical systems. In all our test cases, TRACE is at least as accurate and fast as the hybrid symplectic integrator MERCURIUS. In many cases, TRACE's performance is vastly superior to that of MERCURIUS. In test cases with planet-planet close encounters, TRACE is as accurate as MECURIUS with a 12x speed-up. If close encounters with the central star are considered, TRACE achieves good error performance while MERCURIUS fails to give qualitatively correct results. In ensemble tests of violent scattering systems, TRACE matches the high-accuracy IAS15 while providing a 15x speed-up. In large N systems simulating lunar accretion, TRACE qualitatively gives the same results as IAS15 but at a 41x speed-up. We also discuss some cases such as von Zeipel-Lidov-Kozai cycles where hybrid integrators perform poorly and provide some guidance on which integrator to use for which system. TRACE is freely available within the REBOUND package., Comment: Published in MNRAS. Minor changes

We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal abelian categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional representations of simply-laced quantum loop algebras, we prove the associativity and we establish that the resulting quantization coincides with the quantum Grothendieck ring constructed by Nakajima and Varagnolo-Vasserot in a geometric manner. Hence, it yields a unified representation-theoretic interpretation of the quantum Grothendieck ring. As a second main example, we establish an analogous result for a monoidal category of finite-dimensional modules over symmetric quiver Hecke algebras categorifying the coordinate ring of a unipotent group associated with a Weyl group element. We obtain various applications, in particular on the homological structure of representations., Comment: 61 pages, v3: revised introduction, fixed typos, updated references

Rare event schemes require an approximation of the probability of the rare event as a function of system state. Finding an appropriate reaction coordinate is typically the most challenging aspect of applying a rare event scheme. Here we develop an artificial intelligence (AI) based reaction coordinate that effectively predicts which of a limited number of simulations of the Solar System will go unstable using a convolutional neural network classifier. The performance of the algorithm does not degrade significantly even 3.5 billion years before the instability. We overcome the class imbalance intrinsic to rare event problems using a combination of minority class oversampling, increased minority class weighting, and pulling multiple non-overlapping training sequences from simulations. Our success suggests that AI may provide a promising avenue for developing reaction coordinates without detailed theoretical knowledge of the system., Comment: 3 pages, 1 figure

We present new almost time-reversible integrators for solution of planetary systems consisting of "planets" and a dominant mass ("star"). The algorithms can be considered adaptive generalizations of the Wisdom--Holman method, in which all pairs of planets can be assigned timesteps. These timesteps, along with the global timestep, can be adapted time-reversibly, often at no appreciable additional compute cost, without sacrificing any of the long-term error benefits of the Wisdom--Holman method. The method can also be considered a simpler and more flexible version of the \texttt{SYMBA} symplectic code. We perform tests on several challenging problems with close encounters and find the reversible algorithms are up to $2.6$ times faster than a code based on \texttt{SYMBA}. The codes presented here are available on Github. We also find adapting a global timestep reversibly and discretely must be done in block-synchronized manner or similar., Comment: 11 pages, 7 Figures. Matches accepted MNRAS version

We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories $\mathcal{O}_\mathbb{Z}$ of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called $QQ$-system. We conjecture that all cluster monomials are classes of simple objects of $\mathcal{O}_\mathbb{Z}$. In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply-connected algebraic group. This explains the similarity between $QQ$-system relations and certain generalized minor identities discovered by Fomin and Zelevinsky., Comment: 68 pages, 20 figures. V2 : references added, improved exposition in Sections 10.3, 10.4. V3 : 70 pages. Minor corrections and improvements. Final version to appear in PLMS

Generalized Baxter's TQ-relations and the QQ-system are systems of algebraic relations in the category O of representations of the Borel subalgebra of the quantum affine algebra U_q(g^), which we established in our earlier works arXiv:1308.3444 and arXiv:1606.05301. In the present paper, we conjecture a family of analogous relations labeled by elements of the Weyl group W of g, so that the original relations correspond to the identity element. These relations are closely connected to the W-symmetry of q-characters established in arXiv:2211.09779. We prove these relations for all w in W if g has rank two, and we prove the extended TQ-relations if w is a simple reflection. We also generalize our results and conjectures to the shifted quantum affine algebras., Comment: 42 pages, v3: more details added, accepted for publication in Communications in Math. Phys

Mercury's orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can cause g1 to decrease to the approximately constant value of g5 and create a resonance. Previous work has approximated the variation in g1 as stochastic diffusion, which leads to a phenomological model that can reproduce the Mercury instability statistics of secular and N-body models on timescales longer than 10 Gyr. Here we show that the diffusive model underpredicts the Mercury instability probability by a factor of 3-10,000 on timescales less than 5 Gyr, the remaining lifespan of the Solar System. This is because g1 exhibits larger variations on short timescales than the diffusive model would suggest. To better model the variations on short timescales, we build a new subdiffusive phenomological model for g1. Subdiffusion is similar to diffusion but exhibits larger displacements on short timescales and smaller displacements on long timescales. We choose model parameters based on the behavior of the g1 trajectories in the N-body simulations, leading to a tuned model that can reproduce Mercury instability statistics from 1-40 Gyr. This work motivates fundamental questions in Solar System dynamics: Why does subdiffusion better approximate the variation in g1 than standard diffusion? Why is there an upper bound on g1, but not a lower bound that would prevent it from reaching g5?, Comment: accepted at ApJ

We propose a learning-based method to recover normals, specularity, and roughness from a single diffuse image of a material, using microgeometry appearance as our primary cue. Previous methods that work on single images tend to produce over-smooth outputs with artifacts, operate at limited resolution, or train one model per class with little room for generalization. Previous methods that work on single images tend to produce over-smooth outputs with artifacts, operate at limited resolution, or train one model per class with little room for generalization. In contrast, in this work, we propose a novel capture approach that leverages a generative network with attention and a U-Net discriminator, which shows outstanding performance integrating global information at reduced computational complexity. We showcase the performance of our method with a real dataset of digitized textile materials and show that a commodity flatbed scanner can produce the type of diffuse illumination required as input to our method. Additionally, because the problem might be illposed -- more than a single diffuse image might be needed to disambiguate the specular reflection -- or because the training dataset is not representative enough of the real distribution, we propose a novel framework to quantify the model's confidence about its prediction at test time. Our method is the first one to deal with the problem of modeling uncertainty in material digitization, increasing the trustworthiness of the process and enabling more intelligent strategies for dataset creation, as we demonstrate with an active learning experiment., Comment: CVPR 2023. Project website: https://carlosrodriguezpardo.es/projects/UMat/

We establish a cluster theoretical interpretation of the isomorphisms of [F.-H.-O.-O., J. Reine Angew. Math., 2022] among quantum Grothendieck rings of representations of quantum loop algebras. Consequently, we obtain a quantization of the monoidal categorification theorem of [Kashiwara-Kim-Oh-Park, arXiv:2103.10067]. We establish applications of these new ingredients. First we solve long-standing problems for any non-simply-laced quantum loop algebras: the positivity of $(q,t)$-characters of all simple modules, and the analog of Kazhdan-Lusztig conjecture for all reachable modules (in the cluster monoidal categorification). We also establish the conjectural quantum $T$-systems for the $(q,t)$-characters of Kirillov-Reshetikhin modules. Eventually, we show that our isomorphisms arise from explicit birational transformations of variables, which we call substitution formulas. This reveals new non-trivial relations among $(q, t)$-characters of simple modules., Comment: v1 60 pages; v2 58 pages, the proofs in Section 7 are simplified

We introduce ASSIST, a software package for ephemeris-quality integrations of test particles. ASSIST is an extension of the REBOUND framework and makes use of its IAS15 integrator to integrate test particle trajectories in the field of the Sun, Moon, planets, and 16 massive asteroids, with the positions of the masses coming from the JPL DE441 ephemeris and its associated asteroid perturber file. The package incorporates the most significant gravitational harmonics and general relativistic corrections. ASSIST also accounts for position- and velocity-dependent non-gravitational effects. The first order variational equations are included for all terms to support orbit fitting and covariance mapping. This new framework is meant to provide an open-source package written in a modern language to enable high-precision orbital analysis and science by the small body community. ASSIST is open source, freely distributed under the GNU General Public license, version 3., Comment: Submitted to PSJ (27 January 2023), Revised (28 March 2023)

The VISIONS public survey provides large-scale, multiepoch imaging of five nearby star-forming regions at subarcsecond resolution in the near-infrared. All data collected within the program and provided by the European Southern Observatory (ESO) science archive are processed with a custom end-to-end pipeline infrastructure to provide science-ready images and source catalogs. The data reduction environment has been specifically developed for the purpose of mitigating several shortcomings of the bona fide data products processed with software provided by the Cambridge Astronomical Survey Unit (CASU), such as spatially variable astrometric and photometric biases of up to 100 mas and 0.1 mag, respectively. At the same time, the resolution of coadded images is up to 20% higher compared to the same products from the CASU processing environment. Most pipeline modules are written in Python and make extensive use of C extension libraries for numeric computations, thereby simultaneously providing accessibility, robustness, and high performance. The astrometric calibration is performed relative to the Gaia reference frame, and fluxes are calibrated with respect to the source magnitudes provided in the Two Micron All Sky Survey (2MASS). For bright sources, absolute astrometric errors are typically on the order of 10 to 15 mas and fluxes are determined with subpercent precision. Moreover, the calibration with respect to 2MASS photometry is largely free of color terms. The pipeline produces data that are compliant with the ESO Phase 3 regulations and furthermore provides curated source catalogs that are structured similarly to those provided by the 2MASS survey., Comment: Accepted for publication in Astronomy & Astrophysics on 28 February 2023

VISIONS is an ESO public survey of five nearby (d < 500 pc) star-forming molecular cloud complexes that are canonically associated with the constellations of Chamaeleon, Corona Australis, Lupus, Ophiuchus, and Orion. The survey was carried out with VISTA, using VIRCAM, and collected data in the near-infrared passbands J, H, and Ks. With a total on-sky exposure time of 49.4 h VISIONS covers an area of 650 deg$^2$, and it was designed to build an infrared legacy archive similar to that of 2MASS. Taking place between April 2017 and March 2022, the observations yielded approximately 1.15 million images, which comprise 19 TB of raw data. The observations are grouped into three different subsurveys: The wide subsurvey comprises shallow, large-scale observations and has visited the star-forming complexes six times over the course of its execution. The deep subsurvey of dedicated high-sensitivity observations has collected data on the areas with the largest amounts of dust extinction. The control subsurvey includes observations of areas of low-to-negligible dust extinction. Using this strategy, the VISIONS survey offers multi-epoch position measurements, is able to access deeply embedded objects, and provides a baseline for statistical comparisons and sample completeness. In particular, VISIONS is designed to measure the proper motions of point sources with a precision of 1 mas/yr or better, when complemented with data from VHS. Hence, VISIONS can provide proper motions for sources inaccessible to Gaia. VISIONS will enable addressing a range of topics, including the 3D distribution and motion of embedded stars and the nearby interstellar medium, the identification and characterization of young stellar objects, the formation and evolution of embedded stellar clusters and their initial mass function, as well as the characteristics of interstellar dust and the reddening law., Comment: Accepted for publication in Astronomy & Astrophysics on 19 January 2023

We present a simple algorithm to switch between $N$-body time integrators in a reversible way. We apply it to planetary systems undergoing arbitrarily close encounters and highly eccentric orbits, but the potential applications are broader. Upgrading an ordinary non-reversible switching integrator to a reversible one is straightforward and introduces no appreciable computational burden in our tests. Our method checks if the integrator during the time step violates a time-symmetric selection condition and redoes the step if necessary. In our experiments a few percent of steps would have violated the condition without our corrections. By eliminating them the algorithm avoids long-term error accumulation, of several orders magnitude in some cases., Comment: 10 pages, 8 figures. Matches accepted MNRAS version

The long-term stability of the Solar System is an issue of significant scientific and philosophical interest. The mechanism leading to instability is Mercury's eccentricity being pumped up so high that Mercury either collides with Venus or is scattered into the Sun. Previously, only three five-billion-year $N$-body ensembles of the Solar System with thousands of simulations have been run to assess long-term stability. We generate two additional ensembles, each with 2750 members, and make them publicly available at \texttt{https://archive.org/details/@dorianabbot}. We find that accurate Mercury instability statistics can be obtained by (1) including only the Sun and the 8 planets, (2) using a simple Wisdom-Holman scheme without correctors, (3) using a basic representation of general relativity, and (4) using a time step of 3.16 days. By combining our Solar System ensembles with previous ensembles we form a 9,601-member ensemble of ensembles. In this ensemble of ensembles, the logarithm of the frequency of a Mercury instability event increases linearly with time between 1.3 and 5 Gyr, suggesting that a single mechanism is responsible for Mercury instabilities in this time range and that this mechanism becomes more active as time progresses. Our work provides a robust estimate of Mercury instability statistics over the next five billion years, outlines methodologies that may be useful for exoplanet system investigations, and provides two large ensembles of publicly available Solar System integrations that can serve as testbeds for theoretical ideas as well as training sets for artificial intelligence schemes., Comment: accepted at ApJ

We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of W-invariants of Y is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of U_q(g^). This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category O of representations of the Borel subalgebra of U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616., Comment: 29 pages, v2: a uniform proof of the braid relations for all simple Lie algebras is given in Section 4

Numerical integrations of the Solar System have been carried out for decades. Their results have been used, for example, to determine whether the Solar System is chaotic, whether Mercury's orbit is stable, or to help discern Earth's climate history. We argue that all of the past studies we consider in this work are affected by numerical chaos to different degrees, affecting the possible orbits and instability probability of Mercury, sometimes significantly. We show how to eliminate the effects of numerical chaos by resolving Mercury's pericentre passage. We also show that several higher order symplectic maps do not exhibit significant differences in resolving pericentre passage of Mercury (at fixed time step), making their advantages suspect for calculating long-term orbits. Resolving pericentre passage affects a wide array of orbital numerical studies, like exoplanet studies, studies of the galactic centre, and other $N$-body problems., Comment: 6 pages, 6 figures. Matches accepted MNRAS version