arXiv subjects

Ángel González-Prieto

Publications and source records attributed to Ángel González-Prieto.

At least 19 recordsLinked to original sources

arXiv2026

Root data in character varieties

Given $G$ an algebraic reductive group over an algebraically closed field of characteristic zero and $Γ$ a finitely generated group, we provide a stratification of the $G$-character variety of $Γ$ in terms of conjugacy classes of parabolic subgroups of $G$. Each stratum has the structure of a pseudo-quotient, which is a relaxed GIT notion capturing the topology of the quotient and, therefore, behaving well for motivic computations of invariants of the character varieties. These stratifications are constructed by analyzing the root datum of $G$ to encode parabolic classes. Finally, detailed and explicit motivic formulae are provided for cases with Dynkin diagram of types $A$, $B$, $C$ and $D$.

math.AG↗

arXiv2026

Theory building in software engineering: Operationalization

This work is part of a research project whose ultimate goal is to systematize a procedure for constructing theories in software engineering. The proposed methodology involves four phases: conceptualization, operationalization, testing, and application. In previous work, we described the conceptualization process. This paper presents a set of procedures for systematizing the operationalization phase in theory building. Specifically, it translates the concepts and propositions obtained from the previous conceptualization into constructs and empirically testable hypotheses. The operationalization phase is structured across three distinct dimensions: the practical procedural steps, their strict formal mathematical specification, and their evaluation through an illustrative empirical case study. Grounded in critical realism, we operationalize the qualitatively derived concepts and propositions into explicit constructs and logical hypotheses. It is important to emphasize that a novel strategy has been designed that reduces the number of hypotheses resulting from this process. In the use case described, more than 30% of the initial hypotheses have been eliminated because they are redundant. This paper is a pioneering contribution in offering comprehensive guidelines for theory operationalization using logical implication and describes a strategy based on graph theory that tackles the problem of hypothesis explosion and maintains the principle of parsimony.

cs.SE↗

arXiv2026

Stringy invariants for abelian character varieties

We compute all stringy invariants, encoding orbifold cohomology numbers, of (the normalization of) the identity component of $G$-character varieties of free abelian groups and of moduli spaces of $G$-Higgs bundles on abelian varieties, for all complex connected reductive groups $G$. As an application, we provide a direct proof of a topological mirror symmetry statement: the equality of the stringy invariants for Langlands dual groups. In the framework of Ginzburg--Kaledin local resolutions, we discuss the meaning of these stringy invariants, showing that symplectic resolutions of these moduli spaces can only exist in the cases of groups of Dynkin type $A$, $B$, or $C$. When such resolutions exist, the computed stringy Hodge numbers agree with the actual Hodge numbers of the resolution.

math.AG↗

arXiv2026

Computable functions as Reeb flows

We prove that, given any contact $3$-manifold and any computable function $f: \mathbb{N} \dashrightarrow \mathbb{N}$, there exists a defining contact form and a Poincaré section of its Reeb flow whose partially defined return map computes $f$.

math.SG↗

arXiv2026

Restricted Bernoulli Matrix Factorization: Balancing the trade-off between prediction accuracy and coverage in classification based collaborative filtering

Reliability measures associated with the prediction of the machine learning models are critical to strengthening user confidence in artificial intelligence. Therefore, those models that provide not only predictions, but also reliability, enjoy greater popularity. In the field of recommender systems, reliability is crucial, since users tend to prefer those recommendations that are sure to interest them, that is, high predictions with high reliabilities. In this paper, we propose Restricted Bernoulli Matrix Factorization (ResBeMF), a new algorithm aimed at enhancing the performance of classification-based collaborative filtering. This model is based on a collection of restricted matrix factorizations that jointly generate, for each user-item pair, a full probability distribution over the possible rating scores. To prove its effectiveness, the proposed model has been compared to other existing solutions in the literature in terms of prediction quality (Mean Absolute Error and accuracy scores), prediction quantity (coverage score) and recommendation quality (Mean Average Precision score). The experimental results demonstrate that the proposed model provides a good balance in terms of the quality measures used compared to other recommendation models.

cs.IR↗

arXiv2025

Topological Kleene Field Theories as a model of computation

In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions and drawing parallels with Topological Field Theory. Our central result shows that any computable function can be simulated by the flow on a smooth bordism of a vector field with good local properties, setting an alternative model of computation to Turing machines. We thus establish that a computable function can be fully realized within a single go of a dynamical system, differing from previous works where computation is encoded as an iterative process. The output of the computable function emerges directly, laying the groundwork for potential applications that accelerate the physical realization of computation.

math.DS↗

arXiv2025

Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants

This paper reinterprets Alexander-type invariants of knots via representation varieties of knot groups into the group $\textrm{AGL}_1(\mathbb{C})$ of affine transformations of the complex line. In particular, we prove that the coordinate ring of the $\textrm{AGL}_{1}(\mathbb{C})$-representation variety is isomorphic to the symmetric algebra of the Alexander module. This yields a natural interpretation of the Alexander polynomial as the singular locus of a coherent sheaf over $\mathbb{C}^*$, whose fibres correspond to quandle representation varieties of the knot quandle. As a by-product, we construct Topological Quantum Field Theories that provide effective computational methods and recover the Burau representations of braids. This theory offers a new geometric perspective on classical Alexander invariants and their functorial quantization.

math.GT↗

arXiv2025

Universality in computable dynamical systems: Old and new

The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualizations of computation. Recently, this connection has gained renewed attention, fueled by T. Tao's programme aiming to discover blowing-up solutions of the Navier-Stokes equations using an embedded computational model. In this survey paper, we review some of the recent works that introduce novel and exciting perspectives on the representation of computability through dynamical systems. Starting from dynamical universality in a classical sense, we shall explore the modern notions of Turing universality in fluid dynamics and Topological Kleene Field Theories as a systematic way of representing computable functions by means of dynamical bordisms. Finally, we will discuss some important open problems in the area.

math.DS↗

arXiv2025

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG↗

arXiv2025

Cosymplectic Chern--Hamilton conjecture

In this paper, we study the Chern-Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co-Kähler or if it is a mapping torus of the 2-torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number $b_1 \geq 2$ admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.

math.DG↗

arXiv2025

Virtual Classes of Character Stacks

In this paper, we extend the Topological Quantum Field Theory developed by González-Prieto, Logares, and Muñoz for computing virtual classes of $G$-representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks. To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field Theory is defined. In this way, we compute the virtual class of the character stack over $BG$, that is, a motivic decomposition of the representation variety with respect to the natural adjoint action. We apply this framework in two cases providing explicit expressions for the virtual classes of the character stacks of closed orientable surfaces of arbitrary genus. First, in the case of the affine linear group of rank $1$, the virtual class of the character stack fully remembers the natural adjoint action, in particular, the virtual class of the character variety can be straightforwardly derived. Second, we consider the non-connected group $\mathbb{G}_m \rtimes \mathbb{Z}/2\mathbb{Z}$, and we show how our theory allows us to compute motivic information of the character stacks where the classical naïve point-counting method fails.

math.AG↗

arXiv2024

Neural Collaborative Filtering Classification Model to Obtain Prediction Reliabilities

Neural collaborative filtering is the state of art field in the recommender systems area; it provides some models that obtain accurate predictions and recommendations. These models are regression-based, and they just return rating predictions. This paper proposes the use of a classification-based approach, returning both rating predictions and their reliabilities. The extra information (prediction reliabilities) can be used in a variety of relevant collaborative filtering areas such as detection of shilling attacks, recommendations explanation or navigational tools to show users and items dependences. Additionally, recommendation reliabilities can be gracefully provided to users: "probably you will like this film", "almost certainly you will like this song", etc. This paper provides the proposed neural architecture; it also tests that the quality of its recommendation results is as good as the state of art baselines. Remarkably, individual rating predictions are improved by using the proposed architecture compared to baselines. Experiments have been performed making use of four popular public datasets, showing generalizable quality results. Overall, the proposed architecture improves individual rating predictions quality, maintains recommendation results and opens the doors to a set of relevant collaborative filtering fields.

cs.IR↗

arXiv2024

Incorporating Recklessness to Collaborative Filtering based Recommender Systems

Recommender systems are intrinsically tied to a reliability/coverage dilemma: The more reliable we desire the forecasts, the more conservative the decision will be and thus, the fewer items will be recommended. This causes a detriment to the predictive capability of the system, as it is only able to estimate potential interest in items for which there is a consensus in their evaluation, rather than being able to estimate potential interest in any item. In this paper, we propose the inclusion of a new term in the learning process of matrix factorization-based recommender systems, called recklessness, that takes into account the variance of the output probability distribution of the predicted ratings. In this way, gauging this recklessness measure we can force more spiky output distribution, enabling the control of the risk level desired when making decisions about the reliability of a prediction. Experimental results demonstrate that recklessness not only allows for risk regulation but also improves the quantity and quality of predictions provided by the recommender system.

cs.IR↗

arXiv2024

Character varieties of torus links

In this paper, we study the geometry of the moduli space of representations of the fundamental group of the complement of a torus link into an algebraic group G, an algebraic variety known as the G-character variety of the torus link. These torus links are a family of links in the 3-dimensional sphere formed by stacking several copies of torus knots. We develop an intrinsic stratification of the variety that allows us to relate its geometry with the one of the underlying torus knot. Using this information, we explicitly compute the E-polynomial associated to the Hodge structure of these varieties for $G=SL_2(\mathbb{C})$ and $SL_3(\mathbb{C})$, for an arbitrary torus link, showing an unexpected relation with the number of strands of the link.

math.GT↗

arXiv2024

Representation varieties of twisted Hopf links

We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.

math.GT↗

arXiv2024

Data Augmentation techniques in time series domain: A survey and taxonomy

With the latest advances in Deep Learning-based generative models, it has not taken long to take advantage of their remarkable performance in the area of time series. Deep neural networks used to work with time series heavily depend on the size and consistency of the datasets used in training. These features are not usually abundant in the real world, where they are usually limited and often have constraints that must be guaranteed. Therefore, an effective way to increase the amount of data is by using Data Augmentation techniques, either by adding noise or permutations and by generating new synthetic data. This work systematically reviews the current state-of-the-art in the area to provide an overview of all available algorithms and proposes a taxonomy of the most relevant research. The efficiency of the different variants will be evaluated as a central part of the process, as well as the different metrics to evaluate the performance and the main problems concerning each model will be analysed. The ultimate aim of this study is to provide a summary of the evolution and performance of areas that produce better results to guide future researchers in this field.

cs.LG↗

arXiv2023

Pseudo-quotients of algebraic actions and their application to character varieties

In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of $\mathrm{SL}_{2}(k)$-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type.

math.AG↗

arXiv2023

Arithmetic-Geometric Correspondence of Character Stacks via Topological Quantum Field Theory

In this paper, we introduce Topological Quantum Field Theories (TQFTs) generalizing the arithmetic computations done by Hausel and Rodríguez-Villegas and the geometric construction done by Logares, Muñoz, and Newstead to study cohomological invariants of $G$-representation varieties and $G$-character stacks. We show that these TQFTs are related via a natural transformation that we call the 'arithmetic-geometric correspondence' generalizing the classical formula of Frobenius on the irreducible characters of a finite group. We use this correspondence to extract some information on the character table of finite groups using the geometric TQFT, and vice versa, we greatly simplify the geometric calculations in the case of upper triangular matrices by lifting its irreducible characters to the geometric setting.

math.AG↗