Federated sequential recommendation distributes model training across user devices so that behavioural data remains local, reducing privacy risks. Yet, this setting introduces two intertwined difficulties. On the one hand, individual clients typically contribute only short and highly sparse interaction sequences, limiting the reliability of learned user representations. On the other hand, the federated optimisation process is vulnerable to malicious or corrupted client updates, where poisoned gradients can significantly distort the global model. These challenges are particularly severe in sequential recommendation, where temporal dynamics further complicate signal aggregation. To address this problem, we propose a robust aggregation framework tailored for federated sequential recommendation under sparse and adversarial conditions. Instead of relying on standard averaging, our method introduces a defence-aware aggregation mechanism that identifies and down-weights unreliable client updates while preserving informative signals from sparse but benign participants. The framework incorporates representation-level constraints to stabilise user and item embeddings, preventing poisoned or anomalous contributions from dominating the global parameter space. In addition, we integrate sequence-aware regularisation to maintain temporal coherence in user modelling despite limited local observations.

We report an efficient experimental method for generating high-order Laguerre-Gauss (HOLG) modes by simply coupling a Gaussian beam into the cladding of a multimode fiber (MMF). In particular, the order of the HOLG mode remains invariant with respect to input power, propagation distance, and pulse duration. Furthermore, spectral and power measurements confirm that the beam-shaping mechanism is predominantly linear, whereas Kerr nonlinearity primarily affects the longitudinal phase-matching condition and conversion efficiency, without altering the generated mode order. Altogether, these findings establish our approach as a highly robust and scalable platform for generating tailored optical beams.

Learning-based signal processing systems increasingly support high-stakes medical decisions using heterogeneous biomedical signals, including medical images, physiological time series, and clinical records. Despite strong predictive performance, many models rely on statistical correlations that are unstable across acquisition settings, patient populations, and institutional practices, limiting robustness, interpretability, and clinical trust. We advocate a causal signal processing perspective in which biomedical signals are treated as effects of latent generative mechanisms rather than as isolated predictive inputs. Using clinical risk prediction as a motivating example, we show how disease-related factors generate observable biomarkers, while acquisition processes act as confounders influencing signal appearance. In clinical disease risk prediction from chest CT scans and patient risk factors, correlational models may fail under scanner changes, whereas causal abstractions remain invariant. Building on this view, we propose a unifying conceptual framework integrating causal modeling with learning-based signal processing and neuro-symbolic reasoning. Statistical models extract multimodal representations that are mapped to interpretable causal abstractions and combined with symbolic knowledge encoding clinical risk factors and guidelines. This structure enables clinically grounded explanations, counterfactual reasoning about hypothetical interventions, and improved robustness to distribution shifts arising from changes in acquisition conditions or screening policies. Rather than introducing a specific algorithm, this article presents schematic causal structures and a comparative analysis of correlation-based, causal, and neuro-symbolic approaches to guide the design of robust and interpretable medical decision-support systems.

We present an end-to-end pipeline for large-scale portfolio selection with cardinality constraints and experimentally demonstrate it on trapped-ion quantum processors using hardware-aware decomposition. Building on RMT-based correlation-matrix denoising and community detection, we identify correlated asset groups and introduce a correlation-guided greedy splitting scheme that caps each cluster by the executable qubit budget. Each cluster defines a hardware-embeddable QUBO subproblem that we solve using bias-field digitized counterdiabatic quantum optimization (BF-DCQO), a non-variational method that avoids classical parameter-training loops. We recombine low-energy candidates into global portfolios and enforce feasibility with a two-stage post-processing routine: fast repair followed by a cardinality-preserving swap local search. We benchmark the workflow on a 250-asset universe taken from the S&P 500 and execute subproblems on a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We observe that larger executable subproblem sizes reduce decomposition error and systematically improve final objective values and risk-return trade-offs relative to randomized baselines under identical post-processing. Overall, the results establish a hardware-tested route for scaling financial optimization problems, defined by a trade space in which executable problem size and circuit cost are balanced against the resulting solution quality.

Electronic interferometers have been identified as possible candidates for building electronic flying qubits. Such a regime requires ultrafast voltage pulses whose duration is shorter than the time of flight through the device. Understanding the corresponding physics in the presence of such short excitations requires a proper treatment of electron-electron interactions. In this article, we take a step in this direction by performing time-resolved simulations of a Mach-Zehnder interferometer treating the interactions at the time-dependent mean-field level. We find that the main effect of the interaction is the renormalization of the pulse velocity. Very importantly, the interference effects appear to be robust to the presence of interactions.

Time-reversal symmetry is of fundamental importance to physics. In the classical theory of time-reversal symmetry, the time-reversal symmetry of a quantum system is described by an anti-unitary operator, which is known as the time-reversal operator of the system. In this paper, we introduce and study a model of continuous-time quantum walk on a special locally infinite graph. After examining its spectral property, we investigate the time-reversal symmetry of the model. To our surprise, we find that its time-reversal symmetry can be described directly by a unitary operator, which contrasts sharply with that in the classical theory of time-reversal symmetry. Some other related results are also proven.

Quadratic programming (QP) is a fundamental optimization model with wide-ranging applications in decision-making and machine learning, yet efficiently solving large-scale instances remains a major computational challenge. Building upon the recently developed PDHCG framework, we propose PDHCG-II, an enhanced first-order solver tailored for large-scale convex QPs. The proposed method explicitly exploits the quadratic structure of the objective and incorporates several key algorithmic innovations, including Halpern-type acceleration and a PID-controlled adaptive update of the primal-dual weight. To further improve practical performance, PDHCG-II introduces a refined adaptive termination criterion for inner subproblems to prevent over-solving, together with an infeasibility detection mechanism for robust handling of ill-posed instances. Extensive numerical experiments demonstrate that PDHCG-II consistently achieves 2.5-5 times speedups over PDHCG on standard QP benchmarks. To facilitate reproducibility and broader adoption, we release a CUDA-C implementation of PDHCG-II as open-source software.

In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case of a hypersurface in a projective space, we establish a similar result for very general smooth hypersurfaces of sufficiently large bi-degree. More precisely, we show that the covering gonality of such hypersurfaces can be computed by viewing them as a family of hypersurfaces over a projective space. Then, the curves computing the covering gonality lie entirely within the fibres of one of the families. This rules out transversal curves from computing the covering gonality. In addition to this, we investigate the behaviour of the joint covering gonality as in [LM23] and establish a lower bound for bi-degree large enough.

The transient picture for a Bloch electron accelerating in an arbitrarily time-dependent homogeneous electric field is developed. The temporal sequence for the analysis includes the instant after electron injection, followed by the time required for a small change in electron wavenumber away from initial injection, leading to the final time evolution over many Bloch periods. The time-dependent behavior is studied using the properties of the Schrödinger equation. The electric field is described through the vector potential gauge, and the instantaneous eigenstates of the Bloch, electric-field-dependent Hamiltonian are used as basis states in describing the Bloch dynamics in the electric field. For each temporal sequence considered, the solution to the Schrödinger equation is established and comparatively discussed. The expectation value of the momentum is obtained for the special case of first order in a constant electric field; the resulting velocity derived is a field-dependent generalization of the natural Zitterbewegung-like behavior discussed in the recent literature. The early-time and long-time limits of the momentum expectation value and its time derivative demonstrate that the resistance to Bloch acceleration after initial band injection varies from real mass to effective mass dynamics as the electron accelerates through the band under the influence of electric field. This changing inertia from early injection of a free-mass electron is the result of the {\it real mass} electron {\it dressing-up} into the states of the crystal to become {\it an effective mass} electron. The ramifications of this temporal {\it dressing} behavior are discussed in considering the general dynamics of Bloch electrons subject to ultrastrong electric fields.

Background. Automated test execution is an important activity to gather information about the quality of a software project. So-called flaky tests, however, negatively affect this process. Such tests fail seemingly at random without changes to the code and thus do not provide a clear signal. Previous work proposed to identify flaky tests based on the source code identifiers in the test code. So far, these approaches have not been evaluated in a large-scale industrial setting. Aims. We evaluate approaches to identify flaky tests and their root causes based on source code identifiers in the test code in a large-scale industrial project. Method. First, we replicate previous work by Pinto et al. in the context of SAP HANA. Second, we assess different feature extraction techniques, namely TF-IDF and TF-IDFC-RF. Third, we evaluate CodeBERT and XGBoost as classification models. For a sound comparison, we utilize both the data set from previous work and two data sets from SAP HANA. Results. Our replication shows similar results on the original data set and on one of the SAP HANA data sets. While the original approach yielded an F1-Score of 0.94 on the original data set and 0.92 on the SAP HANA data set, our extensions achieve F1-Scores of 0.96 and 0.99, respectively. The reliance on external data sources is a common root cause for test flakiness in the context of SAP HANA. Conclusions. The vocabulary of a large industrial project seems to be slightly different with respect to the exact terms, but the categories for the terms, such as remote dependencies, are similar to previous empirical findings. However, even with rather large F1-Scores, both finding source code identifiers for flakiness and a black box prediction have limited use in practice as the results are not actionable for developers.

Particle Dobble is an open-access, gamified learning tool designed to address persistent misconceptions in particle physics education by symbolically representing elements of the Standard Model. Aimed at upper secondary and introductory undergraduate students, the card game adapts the combinatorial structure of Dobble to create a deck in which each pair of cards shares exactly one particle symbol, fostering repeated retrieval practice and conceptual discussion. By deliberately replacing literal depictions of particles with abstract icons, the game challenges Newtonian "tiny billiard ball" mental models and promotes a relational, non-classical understanding of elementary particles. Grounded in research on student difficulties in quantum mechanics and gamification in science education, Particle Dobble serves as a low-threshold intervention to prepare learners for more formal engagement with quantum field concepts.

Ramsey-type problems for linear equations began with Schur's theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations $(\mathcal{E}_1,\mathcal{E}_2)$ and determines the minimum integer $N$ for which any red-blue coloring of $\{1,2,...,N\}$ forces either a red solution of the equation $\mathcal{E}_1$ or a blue solution of the equation $\mathcal{E}_2$. In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms $x+y+c=z$ and $x+qy=z$. We determine the exact two-color off-diagonal Rado number $R_2(c,q)$ associated with this system of equations.

Natural Language Processing (NLP) provides highly effective tools for interpreting and handling human language, offering a broad spectrum of applications. In this paper, we address a classic combinatorial problem -- finding graph partitions with high modularity -- by applying NLP techniques that compute term frequency and inverse document frequency (TF-IDF) alongside machine learning clustering algorithms. We present a new framework, called Clique-TF-IDF, designed for graph partitioning, a task that holds significant relevance across various network analysis contexts. This approach uses dense substructures of the graph, specifically maximal cliques, to represent each vertex in terms of the cliques it is part of, in a manner akin to term-document matrices. Experiments show that Clique-TF-IDF yields results that are comparable to or outperform the current state-of-the-art algorithms, whether or not the number of partitions is known in advance. Although this framework emphasizes on cliques and partitioning, it can be extended to devise AI-driven solutions for a variety of challenging combinatorial problems that can leverage efficiently enumerable substructures.

Hypercomplex signal processing (HSP) offers powerful tools for analyzing and processing multidimensional signals by explicitly exploiting inter-dimensional correlations through Clifford algebra. In recent years, hypercomplex formulations of the phase retrieval (PR) problem, wheren a complex-valued signal is recovered from intensity-only measurements, have attracted growing interest. Hypercomplex phase retrieval (HPR) naturally arises in a range of optical imaging and computational sensing applications, where signals are often modeled using quaternion- or octonion-valued representations. Similar to classical PR, HPR problems may involve measurements obtained via complex, hypercomplex, Fourier, or other structured sensing operators. These formulations open new avenues for the development of advanced HSP-based algorithms and theoretical frameworks. This chapter surveys emerging methodologies and applications of HPR, with particular emphasis on optical imaging systems.

We propose a causal predictive framework for estimating risk under preventative interventions. The Unexposed Mediator Model maintains mediators that are also predictors at their unexposed level, removing double counting of intervention effects at followup visits. The Modifiable Risk Factor Model handles multiple interventions flexibly by modelling their effects via mediators that are also predictors, assuming a known causal structure. The Two Component Model combines a predictive baseline model with an intervention model to improve predictive performance. We illustrate the framework in primary prevention of cardiovascular disease. The proposed models allow arbitrary interventions to be evaluated within a prediction under intervention framework, with causally consistent risk estimates across repeated visits. Limitations include reliance on predictor values from an arbitrary first visit, requirements for causal structural knowledge, and a consistency assumption, that interventions with identical effects on predictors have identical effects on outcomes, which warrant further investigation.

We work towards a question raised by Cluckers and Glazer in [CG25], to bring the dimension growth upper bounds and lower bounds for the worst case closer together. To this end, we introduce a sublinear sharpened version of the projective Schwartz-Zippel bound. We prove several cases, including the case of configurations of linear varieties. This leads to subquadratic dimension growth bounds in some low dimensions, improving on the quadratic dependence obtained by Binyamini, Cluckers and Kato in [BCK25]. We introduce a natural projection argument with pull-backs and use this to address a second question by Cluckers and Glazer by extending the quadratic dimension growth bounds from [BCK25] to arbitrary codimension.

For any acyclic quiver, Keller-Scherotzke provided a stratifying functor from the category of finite-dimensional modules of the singular Nakajima category to the derived category of the quiver. Under this functor, a degeneration of strata of a graded quiver variety corresponds to a degeneration, in the sense of Jensen-Su-Zimmermann, in the derived category. In this article, for any Dynkin quiver, we further investigate Jensen-Su-Zimmermann's partial order and show that any degeneration of objects in the derived category can be obtained in this way.

The global Jacquet--Langlands correspondence is an instance of Langlands functoriality, namely the expected lifting of the irreducible automorphic representations of an inner form of the general linear group to the split form via the identity morphism of $L$-groups. It is established, by the work of Badulescu, in the case of irreducible components of the discrete spectrum. The purpose of this paper is to extend this correspondence beyond the discrete spectrum. To this end, the point of view of the Franke filtration of spaces of automorphic forms is taken. In fact, our technical key ingredient is a functorial refinement of the Franke filtration, which allows us to establish the Jacquet--Langlands correspondence between consecutive quotients of this refined filtration on the general linear group and its inner form. As a result, our extended Jacquet--Langlands correspondence properly extends Badulescu's correspondence and contains the full functorial lift, predicted by Langlands functoriality.

Thin-film dewetting is classically governed by an adhesion-mediated spinodal instability in which curvature-driven diffusion controls post-rupture coarsening. We show that internal activity fundamentally restructures this instability. Using a minimal microscopic model of an active liquid film on a solid substrate, we identify a competition between active stresses and film-substrate adhesion that produces two independently regulated dynamical length scales: vertical liquid accumulation and lateral rupture propagation. While passive films exhibit universal diffusion-limited growth, $\ell_z(t)\sim t^{1/3}$, activity converts transport from curvature-controlled diffusion to persistence-driven motion, yielding a continuous increase of the coarsening exponent from $\approx 0.33$ to $\approx 0.6$. The growth law analysis shows that persistent self-propulsion introduces an advective flux that competes with curvature-induced chemical potential gradients, enhancing growth when the persistence length becomes comparable to the evolving domain size. Simultaneously, the rupture front transitions from dissipative spreading to strongly accelerated propagation approaching ballistic scaling. This decoupling shows that activity does not simply renormalize effective surface forces but generates a distinct nonequilibrium interfacial instability governed by the balance between persistence length and adhesion. The results provide a minimal physical mechanism linking classical thin-film dewetting to dewetting-like rupture observed in active and biological materials.

This paper investigates the structural identifiability and a comparative calibration of four water retention curves (WRCs) within the framework of the Richards equation coupled with Darcy's law for capillary imbibition in porous media. The considered models -- two classical physically-based laws and two abstract parametrisations developed for building stones -- are consistently reformulated by expressing the hydraulic conductivity $K(Θ)$ and capillary pressure $ψ(Θ)$ independently, allowing the nonlinear diffusion coefficient $D(Θ)$ to be reconstructed in a unified structural form. This common representation enables a rigorous mathematical comparison across models with different theoretical foundations. All models are calibrated against the same experimental imbibition dataset using a grid-based optimisation strategy with adaptive refinement. The analysis reveals a structural property of the associated inverse problem: the hydraulic conductivity and the capillary pressure scale enter the governing equation multiplicatively and therefore cannot be independently identified from imbibition data. Only their product acts as an observable diffusion parameter, where model discrimination is primarily governed by the shape of the resulting effective diffusion function. To the best of our knowledge, this is the first study providing a coherent cross-calibration of these WRCs against an identical dataset within a unified computational framework. Our open-source implementation, released within the Stoneverse platform, provides a reproducible baseline for further developments, including probabilistic inversion and learning-based approaches.

Ultracold molecules provide opportunities for exploring quantum matter, chemical dynamics and information processing thanks to their rich interactions, which can be controlled by external fields. Magnetic fields tune interactions through Feshbach resonances, enabling the formation of ultracold dimers and triatomic molecules from atom-dimer collisions. Here we demonstrate electric-field control of atom-molecule Feshbach resonances. In mixtures of ground-state sodium-potassium molecules and potassium atoms, electric fields shift resonance positions systematically, revealing specific trimer bound states and their electric-field dependent energies. The response differs markedly from isolated dimers, showing hindered rotation of the molecular constituent near an atom. Electric fields therefore add an independent knob for atom-molecule resonances, open spectroscopic access to triatomic quantum states, and advance controlled polyatomic quantum matter.

We present a multiparty session type (MST) framework with asynchronous mixed choice (MC). We propose a core construct for MC that allows transient inconsistencies in protocol state between distributed participants, but ensures all participants can always eventually reach a mutually consistent state. We prove the correctness of our system by establishing a progress property and an operational correspondence between global types and distributed local type projections. Based on our theory, we implement a practical toolchain for specifying and validating asynchronous MST protocols featuring MC, and programming compliant gen_statem processes in Erlang/OTP. We test our framework by using our toolchain to specify and reimplement part of the amqp_client of the RabbitMQ broker for Erlang.

Microservice architectures are an emergent technology that builds business logic into a suite of small services. Each microservice runs in its process and the communication is made through lightweight mechanisms, usually HTTP resource API. These architectures are built upon independently deployable and, supposedly, reliable pieces of software that may, or may not, have been developed by the team using it. Nowadays, industries are dangerously migrating into microservice architectures without an effective and automatic process for testing the software being used. Furthermore, current API specification languages are not expressive enough to be used for testing purposes. To solve this problem it is necessary to extend currently broadly used API specification languages. APOSTL is a specification language to annotate APIs specifications based on first-order logic, with some restrictions. It has the purpose of extending the currently used API description languages with properties that can be useful for testing purposes, transforming these description documents into useful testing artefacts. Besides providing information needed for testing an application, APOSTL also provides an API with semantic. This additional information is then leveraged to automate microservice testing. The work developed in this thesis aims to fully automate the microservice testing process. It is achieved by the implementation of PETIT a tool able to test microservices when provided with an OpenAPI Specification document, written in JSON and properly annotated with the previously proposed specification language, APOSTL. The tool is able to analyze microservices independently from the source code availability.

This article reports the outcomes of the FAIRNESS COST Action (CA20108), a coordinated European initiative aimed at advancing micrometeorological data toward compliance with the FAIR (Findable, Accessible, Interoperable, Reusable) principles. The article presents three core achievements: (i) a structured inventory of urban and rural micrometeorological networks across Europe; (ii) the design and deployment of the FAIR Micrometeorological Portal, providing a digital infrastructure for data discovery, access, and standardized metadata description; and (iii) methodological guidance for quality control, gap detection, and gap filling tailored to the specific characteristics of micrometeorological time series. By providing both technical infrastructure and community-driven standards, the FAIRNESS outputs advance micrometeorological data from isolated datasets into coherent, reusable resources. Beyond technical developments, the FAIRNESS systematically addressed gaps in knowledge and skills within the micrometeorological community. A key outcome is the beginner-oriented book Micrometeorological Measurements - An Introduction for Beginners, which provides structured guidance on measurement design, instrumentation, data management, and quality assurance. In parallel, FAIRNESS implemented a comprehensive capacity-building programme, including summer schools, workshops, and short-term scientific missions, targeting both domain-specific competencies and transferable skills such as FAIR data stewardship, interdisciplinary collaboration, and practical problem solving. Together, these efforts contribute to strengthening the long-term usability of micrometeorological data and fostering a more integrated, FAIR-oriented research culture within the European meteorological community.

Why do some speakers capture a room almost instantly while others fail to connect? The real-time architecture of audience engagement remains largely a black box. Here, we used motion-captured animations to present the pure nonverbal performance of public speakers to audiences - either in silence (nonverbal-only) or paired with the verbal content (nonverbal-plus-verbal). Using continuous response measurement (CRM), we find that audience judgments solidify with remarkable speed: Moment-to-moment engagement ratings become highly predictive of subsequent evaluations within the initial 10 seconds of the performance. Most notably, this predictive relationship emerged faster and slightly stronger in the nonverbal-only condition, with predictive information being present already after less than 5 seconds. These findings elucidate the social impact a speaker's nonverbal performance has on audience impressions, even when dissociated from the verbal content of the speech. Our approach provides a high-resolution temporal map of social impression formation, pointing to an early "moment of capture" that appears to set the stage for the reception of the following message. On a broader scale, this research validates a powerful new method to isolate different communicative channels, to scientifically deconstruct rhetorical skill, and to study the pervasive impact of nonverbal behavior more broadly. It also enables us to translate the ancient art of rhetoric into a modern science of social impression formation, yielding an empirical basis that can inform human-centered feedback, develop AI-based augmentation tools, and guide the design of engaging, socially present avatars in an increasingly AI-mediated and virtual world.

In this paper we prove global regularity results and Schauder estimates for non-divergence stationary operators of the form L=\sum_{i,j=1}^m a_{ij}(x) X_i X_j, where X_1, ..., X_m are homogeneous (but not necessarily left-invariant) Hörmander vector fields in R^n (n>m), and [a_{ij}(x)] is a symmetric uniformly positive-definite matrix with Hölder-continuous entries w.r.t. the control distance induced by the vector fields X_1, ..., X_m.

In an era of ubiquitous data collection, platform dominance, and AI-mediated governance, the social contract of digital life is increasingly shaped by a few private actors rather than democratic deliberation. This paper advances a dignity-centric Digital Social Contract grounded in data sovereignty, human dignity, and data personalism: the view that personal data are rights-laden emanations of the person and should be protected as a human right, not treated as neutral inputs or tradable commodities. Drawing on social contract theory and interdisciplinary scholarship across law, ethics, economics, computer science, sociology, and political philosophy, we diagnose how datafied infrastructures and surveillance-based business models convert everyday traces into profiles, predictions, and consequential decisions at scale, concentrating informational power and weakening consent, autonomy, and civic trust. We contrast DatAIsm (an extractive paradigm reducing persons to datapoints, optimizing for prediction and control) with HumAIsm, which recenters the human subject and the irreducibility of dignity to mere calculation. We then articulate a governance architecture around six dimensions: (1) technological oversight through Dignity-by-Design, (2) limits to automation and meaningful human control, (3) contextual valuation, redistribution, and incentives, (4) political-institutional legitimacy and multi-actor governance, (5) sociocultural cohesion and the digital commons, and (6) legal-regulatory guarantees. The framework is operationalized through auditable tools (principles, non-negotiable limits, and DbD checklists) aimed at aligning innovation with autonomy, equality, and human flourishing. We conclude by articulating open questions and tensions to foster interdisciplinary debate and guide future research.

Let D be a quaternion division algebra over a non-archimedean local field F of characteristic zero. This article demonstrates the existence and uniqueness of the symplectic model for a family of Zelevinsky modules of GL(n, D) to a family of irreducible representations of GL(n, D). For this family of irreducible representations, we identify a necessary condition under which a symplectic model can exist. This work extends a result of Offen and Sayag beyond the case D = F.

Neutrino non-standard interactions (NSI) with electrons, predicted in many extended theoretical models of particle physics, are known to alter the picture of neutrino decoupling from the cosmic plasma. We update previous analyses of neutrino decoupling in presence of NSI with electrons, extending the parameter space in order to provide, for the first time, a full study of their effect on the production of light elements during Big Bang Nucleosynthesis (BBN). We compare the BBN bounds on non-universal and flavour-changing NSI parameters with the constraints from terrestrial experiments. Our results show that the limits from BBN are significantly less stringent than the experimental bounds, but they are complementary and can provide a test of neutrino physics at different temperature scales and epochs.

This Review is devoted to the presentation of the exact factorization as a framework employed to study a variety of quantum-mechanical many-body problems. Since its original formulation in the 70s, the main applications of the exact factorization were directed towards understanding the properties of multi-component systems, e.g. electron-nuclear systems. Especially in the electron-nuclear case, the exact factorization is viewed as an exactification of the Born-Oppenheimer approximation, thus it was, and still is, largely employed in nonadiabatic dynamics. Nonetheless, as early as the 80s, the formalism was employed to study many-electron interacting systems and, quite recently, i.e. less than a decade ago, it was extended to study the behavior of molecules in the context in cavity quantum electrodynamics. These formulations, perhaps less popular than the electron-nuclear formulation, have attracted a lot of attention over the years. Therefore, we review here the exact electron-only factorization and the exact photon-electron-nuclear factorization.