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Under some technical assumptions, and building on joint work with Bezrukavnikov, we prove a multiplicity formula for indecomposable tilting perverse sheaves on affine flag varieties, with coefficients in a field of characteristic $p$, in terms of $p$-Kazhdan--Lusztig polynomials. Under the same assumptions, we also explain the construction of a "degrading functor" relating mixed modular perverse sheaves (as defined in joint work with Achar) on such varieties to ordinary perverse sheaves., Comment: 34 pages. arXiv admin note: substantial text overlap with arXiv:2206.03738

This paper explores the potential of AI-powered tools to reshape data analysis, focusing on design considerations and challenges. We explore how the emergence of large language and multimodal models offers new opportunities to enhance various stages of data analysis workflow by translating high-level user intentions into executable code, charts, and insights. We then examine human-centered design principles that facilitate intuitive interactions, build user trust, and streamline the AI-assisted analysis workflow across multiple apps. Finally, we discuss the research challenges that impede the development of these AI-based systems such as enhancing model capabilities, evaluating and benchmarking, and understanding end-user needs.

Multi-fidelity surrogate models combining dimensionality reduction and an intermediate surrogate in the reduced space allow a cost-effective emulation of simulators with functional outputs. The surrogate is an input-output mapping learned from a limited number of simulator evaluations. This computational efficiency makes surrogates commonly used for many-query tasks. Diverse methods for building them have been proposed in the literature, but they have only been partially compared. This paper introduces a unified framework encompassing the different surrogate families, followed by a methodological comparison and the exposition of practical considerations. More than a dozen of existing multi-fidelity surrogates have been implemented under the unified framework and evaluated on a set of benchmark problems. Based on the results, guidelines and recommendations are proposed regarding multi-fidelity surrogates with functional outputs. Our study shows that most multi-fidelity surrogates outperform their tested single-fidelity counterparts under the considered settings. But no particular surrogate is performing better on every test case. Therefore, the selection of a surrogate should consider the specific properties of the emulated functions, in particular the correlation between the low- and high-fidelity simulators, the size of the training set, the local nonlinear variations in the residual fields, and the size of the training datasets.

We extend the ramified geometric Satake equivalence due to Zhu (for tamely ramified groups) and the third named author (in full generality) from rational coefficients to include modular and integral coefficients., Comment: 79 pages

These notes present an application of the geometric Satake equivalence to the description of characters of indecomposable tilting modules for reductive algebraic groups over fields of positive characteristic, obtained in joint work with G. Williamson., Comment: 43 pages. Comments welcome!

In this paper we construct equivalences of monoidal categories relating three geometric or representation-theoretic categorical incarnations of the affine Hecke algebra of a connected reductive algebraic group $G$ over a field of positive characteristic: a category of Harish-Chandra bimodules for the Lie algebra of $G$; the derived category of equivariant coherent sheaves on (a completed version of) the Steinberg variety of the Frobenius twist $G^{(1)}$ of $G$; a derived category of constructible sheaves on the affine flag variety of reductive group which is Langlands dual to $G^{(1)}$. These constructions build on the localization theory developed by the first author with Mirkovi\'c and Rumynin and previous work of ours (partly joint with L. Rider), and provide an analogue for positive-characteristic coefficients of a construction of the first author. As an application, we prove a conjecture by Finkelberg-Mirkovi\'c giving a geometric realization of the principal block of algebraic representations of $G$., Comment: v1: 107 pages; v2: 110 pages, minor revision

We review the construction (due to Brenner--Schr\"oer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for $\mathbb{N}$-graded rings; we extend some classical results (in particular, regarding quasi-coherent sheaves on such schemes) from the $\mathbb{N}$-graded setting to this general setting, and prove new results that make sense only in the general setting of Brenner--Schr\"oer. Finally, we show that flag varieties of reductive groups, as well as some vector bundles over such varieties attached to representations of a Borel subgroup, can be naturally interpreted in this formalism.

We unveil a novel mechanism for quasicondensation of hard-core bosons in the presence of quasiperiodicity-induced multifractal single-particle states. The new critical state, here dubbed fractal quasicondensate, is characterized by natural orbitals with multifractal properties and by an occupancy of the lowest natural orbital, {\lambda}0 ~ L{\gamma}, which grows with system size but with a nonuniversal scaling exponent, {\gamma} < 1/2. In contrast to fractal quasicondensates obtained when the chemical potential lies in a region of multifractal single-particle states, placing the chemical potential in regions of localized or delocalized states yields, respectively, no condensation or the usual 1D quasicondensation with {\gamma} = 1/2. Our findings are established by studying one-dimensional hardcore bosons subjected to various quasiperiodic potentials, including the well-known Aubry-Andre model, employing a mapping to non-interacting fermionics that allows for numerically exact results. We discuss how to test our findings in state-of-the-art ultracold atom experiments., Comment: 8 pages (SM included)

We present Hanstreamer, a free and open-source system for webcam-based data presentation. The system performs real-time gesture recognition on the user's webcam video stream to provide interactive data visuals. Apart from the standard chart and map visuals, Hanstreamer is the first such video data presentation system to support network visualisation and interactive DimpVis-style time-series data exploration. The system is ready for use with popular online meeting software such as Zoom and Microsoft Teams., Comment: 3 pages, 5 figures

The investigation of mixture models is a key to understand and visualize the distribution of multivariate data. Most mixture models approaches are based on likelihoods, and are not adapted to distribution with finite support or without a well-defined density function. This study proposes the Augmented Quantization method, which is a reformulation of the classical quantization problem but which uses the p-Wasserstein distance. This metric can be computed in very general distribution spaces, in particular with varying supports. The clustering interpretation of quantization is revisited in a more general framework. The performance of Augmented Quantization is first demonstrated through analytical toy problems. Subsequently, it is applied to a practical case study involving river flooding, wherein mixtures of Dirac and Uniform distributions are built in the input space, enabling the identification of the most influential variables., Comment: 18 figures, 43 pages. Submitted to Statistics and Computing

Quantization summarizes continuous distributions by calculating a discrete approximation. Among the widely adopted methods for data quantization is Lloyd's algorithm, which partitions the space into Vorono\"i cells, that can be seen as clusters, and constructs a discrete distribution based on their centroids and probabilistic masses. Lloyd's algorithm estimates the optimal centroids in a minimal expected distance sense, but this approach poses significant challenges in scenarios where data evaluation is costly, and relates to rare events. Then, the single cluster associated to no event takes the majority of the probability mass. In this context, a metamodel is required and adapted sampling methods are necessary to increase the precision of the computations on the rare clusters., Comment: 7 pages, 4 figures. Submitted to Journal Of Open Source Software

EXplainable Artificial Intelligence (XAI) aims to help users to grasp the reasoning behind the predictions of an Artificial Intelligence (AI) system. Many XAI approaches have emerged in recent years. Consequently, a subfield related to the evaluation of XAI methods has gained considerable attention, with the aim to determine which methods provide the best explanation using various approaches and criteria. However, the literature lacks a comparison of the evaluation metrics themselves, that one can use to evaluate XAI methods. This work aims to fill this gap by comparing 14 different metrics when applied to nine state-of-the-art XAI methods and three dummy methods (e.g., random saliency maps) used as references. Experimental results show which of these metrics produces highly correlated results, indicating potential redundancy. We also demonstrate the significant impact of varying the baseline hyperparameter on the evaluation metric values. Finally, we use dummy methods to assess the reliability of metrics in terms of ranking, pointing out their limitations., Comment: 16 pages, 7 figures

In this paper we generalize and study a notion of (unipotent) nearby cycles over a higher dimensional base based on Be\u{i}linson's description of unipotent nearby cycles, following an idea of Gaitsgory. This generalization, in the setting of affine Grassmannians, is required in recent work of Bezrukavnikov-Braverman-Finkelberg-Kazhdan., Comment: 33 pages. v2: added lots of details, especially on \'etale sheaves

We demonstrate that quasiperiodicity can radically change the ground state properties of 1D moir\'e systems with respect to their periodic counterparts. By studying an illustrative example we show that while narrow bands play a significant role in enhancing interactions both for commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the quasiperiodic-enabled state has contributions from infinitely many wave vectors. This quasi-factal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moir\'e structures and signal out multifractal non-interacting phases, recently found in 2D incommensurate moir\'e systems, as particularly promising parent states.

It is likely that AI systems driven by pre-trained language models (PLMs) will increasingly be used to assist humans in high-stakes interactions with other agents, such as negotiation or conflict resolution. Consistent with the goals of Cooperative AI \citep{dafoe_open_2020}, we wish to understand and shape the multi-agent behaviors of PLMs in a pro-social manner. An important first step is the evaluation of model behaviour across diverse cooperation problems. Since desired behaviour in an interaction depends upon precise game-theoretic structure, we focus on generating scenarios with particular structures with both crowdworkers and a language model. Our work proceeds as follows. First, we discuss key methodological issues in the generation of scenarios corresponding to particular game-theoretic structures. Second, we employ both crowdworkers and a language model to generate such scenarios. We find that the quality of generations tends to be mediocre in both cases. We additionally get both crowdworkers and a language model to judge whether given scenarios align with their intended game-theoretic structure, finding mixed results depending on the game. Third, we provide a dataset of scenario based on our data generated. We provide both quantitative and qualitative evaluations of UnifiedQA and GPT-3 on this dataset. We find that instruct-tuned models tend to act in a way that could be perceived as cooperative when scaled up, while other models seemed to have flat scaling trends.

We construct a "Koszul duality" equivalence relating the (diagrammatic) Hecke category attached to a Coxeter system and a given realization to the Hecke category attached to the same Coxeter system and the dual realization. This extends a construction of Beilinson-Ginzburg-Soergel and Bezrukavnikov-Yun in a geometric context, and of the first author with Achar, Makisumi and Williamson. As an application, we show that the combinatorics of the "tilting perverse sheaves" considered in arXiv:1802.07651 is encoded in the combinatorics of the canonical basis of the Hecke algebra of $(W,S)$ attached to the dual realization., Comment: v2: We corrected an issue with gradings in Section 5 by introducing properly the karoubian closure of the category FM(h,W) in Section 4.8. This is the final version accepted in Annals of Representation Theory

In this paper we determine the scheme-theoretic fixed points of pinned reductive group schemes acted upon by a group of pinning-preserving automorphisms. The results are used in a companion paper to establish a ramified geometric Satake equivalence with integral or modular coefficients., Comment: Comments welcome!

Visualization is an essential operation when assessing the risk of rare events such as coastal or river floodings. The goal is to display a few prototype events that best represent the probability law of the observed phenomenon, a task known as quantization. It becomes a challenge when data is expensive to generate and critical events are scarce, like extreme natural hazard. In the case of floodings, each event relies on an expensive-to-evaluate hydraulic simulator which takes as inputs offshore meteo-oceanic conditions and dyke breach parameters to compute the water level map. In this article, Lloyd's algorithm, which classically serves to quantize data, is adapted to the context of rare and costly-to-observe events. Low probability is treated through importance sampling, while Functional Principal Component Analysis combined with a Gaussian process deal with the costly hydraulic simulations. The calculated prototype maps represent the probability distribution of the flooding events in a minimal expected distance sense, and each is associated to a probability mass. The method is first validated using a 2D analytical model and then applied to a real coastal flooding scenario. The two sources of error, the metamodel and the importance sampling, are evaluated to quantify the precision of the method., Comment: 40 pages, 11 Figures, published in Journal of Computational and Graphical Statistic

In this paper we provide a "combinatorial" description of the category of tilting perverse sheaves on the affine flag variety of a reductive algebraic group, and its free-monodromic variant, with coefficients in a field of positive characteristic. This provides a replacement for the familiar "Soergel theory" for characteristic-0 coefficients, and the second step in our project towards the construction of an equivalence of categories relating the two natural geometric realizations of the associated affine Hecke algebra in the case of positive-characteristic coefficients., Comment: v1: 164 pages, this paper is a sequel to arXiv:2005.05583 (whose title will be updated); v2: 148 pages, minor revision, some changes of notation for consistency with later paper arXiv:2402.08281, removed Section 13 (which will be treated in a separate paper)

We consider chance constrained optimization where it is sought to optimize a function while complying with constraints, both of which are affected by uncertainties. The high computational cost of realistic simulations strongly limits the number of evaluations and makes this type of problems particularly challenging. In such a context, it is common to rely on Bayesian optimization algorithms. Assuming, without loss of generality, that the uncertainty comes from some of the inputs, it becomes possible to build a Gaussian process model in the joint space of design and uncertain variables. A two-step acquisition function is then used to provide, both, promising optimization variables associated to relevant uncertain samples. Our overall contribution is to correlate the constraints in the GP model and exploit this to optimally decide, at each iteration, which constraint should be evaluated and at which point. The coupled Gaussian model of the constraints relies on an output-as-input encoding. The constraint selection idea is developed by enabling that each constraint can be evaluated for a different uncertain input, thus improving the refinement efficiency. Constraints coupling and selection are gradually implement in 3 algorithm variants which are compared to a reference Bayesian approach. The results are promising in terms of convergence speed, accuracy and stability as observed on a 2, a 4 and a 27-dimensional problems.

Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group $G$ over a field of positive characteristic, and following an idea of Arkhipov--Bezrukavnikov--Braverman--Gaitsgory--Mirkovi\'c, we define and initiate the study of some categories of perverse sheaves on the affine Grassmannian of the Langlands dual group to $G$ that should provide geometric models for blocks of representations of the Frobenius kernel $G_1$ of $G$. In particular, we show that these categories admit enough projective and injective objects, which are closely related to some tilting perverse sheaves, and that they are highest weight categories in an appropriate generalized sense., Comment: 89 pages. v2: minor corrections and updates

Cooperation in settings where agents have both common and conflicting interests (mixed-motive environments) has recently received considerable attention in multi-agent learning. However, the mixed-motive environments typically studied have a single cooperative outcome on which all agents can agree. Many real-world multi-agent environments are instead bargaining problems (BPs): they have several Pareto-optimal payoff profiles over which agents have conflicting preferences. We argue that typical cooperation-inducing learning algorithms fail to cooperate in BPs when there is room for normative disagreement resulting in the existence of multiple competing cooperative equilibria, and illustrate this problem empirically. To remedy the issue, we introduce the notion of norm-adaptive policies. Norm-adaptive policies are capable of behaving according to different norms in different circumstances, creating opportunities for resolving normative disagreement. We develop a class of norm-adaptive policies and show in experiments that these significantly increase cooperation. However, norm-adaptiveness cannot address residual bargaining failure arising from a fundamental tradeoff between exploitability and cooperative robustness., Comment: Accepted at the Cooperative AI workshop and the Strategic ML workshop at NeurIPS 2021

Most real optimization problems are defined over a mixed search space where the variables are both discrete and continuous. In engineering applications, the objective function is typically calculated with a numerically costly black-box simulation.General mixed and costly optimization problems are therefore of a great practical interest, yet their resolution remains in a large part an open scientific question. In this article, costly mixed problems are approached through Gaussian processes where the discrete variables are relaxed into continuous latent variables. The continuous space is more easily harvested by classical Bayesian optimization techniques than a mixed space would. Discrete variables are recovered either subsequently to the continuous optimization, or simultaneously with an additional continuous-discrete compatibility constraint that is handled with augmented Lagrangians. Several possible implementations of such Bayesian mixed optimizers are compared. In particular, the reformulation of the problem with continuous latent variables is put in competition with searches working directly in the mixed space. Among the algorithms involving latent variables and an augmented Lagrangian, a particular attention is devoted to the Lagrange multipliers for which a local and a global estimation techniques are studied. The comparisons are based on the repeated optimization of three analytical functions and a beam design problem., Comment: Accepted for publication in Advanced Modeling and Simulation in Engineering Sciences, march 2022

We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg-Mirkovi\'c conjecture) of the Steinberg tensor product formula for simple representations of reductive groups over fields of positive characteristic., Comment: 25 pages. v2: minor revisions