# Ángel González-PrietoComplutense University of Madrid | UCM · Departamento de Álgebra Geometría y Topología

Ángel González-Prieto

PhD in Mathematics

## About

73

Publications

8,311

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577

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Introduction

My research lies in the interface between complex geometry, algebraic geometry and theoretical physics. I am especially focused on Topological Quantum Field Theories, Geometric Invariant Theory, representation theory and Hodge theory. Moreover, I am interested in algebraic topology, especially in higher category theory and functor calculus.
As a byproduct, I am interested in moduli spaces, mainly moduli spaces of parabolic Higgs bundles, and their relation with character varieties, gauge theory and theoretical physics.
Finally, I also work in theoretical Machine Learning oriented to recommendation systems and manifold learning.

Additional affiliations

Education

September 2014 - September 2015

September 2009 - May 2014

## Publications

Publications (73)

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 thei...

Given $G$ an algebraic reductive group over an algebraically closed field of characteristic zero and $\Gamma$ a finitely generated group, we provide a stratification of the $G$-character variety of $\Gamma$ 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...

We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with n n twists, namely Γ n = ⟨ x , y | [ x n , y ] = 1 ⟩ {\Gamma }_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle into the group S U ( r ) SU(r) . For arbitrary rank, we provide geometric descriptions of the loci of irreducible...

Recommender systems that include some reliability measure of their predictions tend to be more conservative in forecasting, due to their constraint to preserve reliability. This leads to a significant drop in the coverage and novelty that these systems can provide. In this paper, we propose the inclusion of a new term in the learning process of mat...

In recent times, recommender systems (RSs) have been attracting a lot of attention from the research community because of their groundbreaking applications [...]

In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of G-representation varieties over manifolds with conic singularities, which we call nodefolds. This construction is valid for any algebraic group G, in any dimension and also in the parabolic s...

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 wor...

Unsupervised machine learning lacks ground truth by definition. This poses a major difficulty when designing metrics to evaluate the performance of such algorithms. In sharp contrast with supervised learning, for which plenty of quality metrics have been studied in the literature, in the field of dimensionality reduction only a few over-simplistic...

We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $\Gamma_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representatio...

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 Geometric Invariant Theory (GIT) quotients regardless of their algebraic properties. The flexibility granted by their topological nature e...

In this paper, 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 SLr(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgr...

Deep learning provides accurate collaborative filtering models to improve recommender system results. Deep matrix factorization and their related collaborative neural networks are the state of the art in the field; nevertheless, both models lack the necessary stochasticity to create the robust, continuous, and structured latent spaces that variatio...

Gender-based crime is one of the most concerning scourges of contemporary society, and governments worldwide have invested lots of economic and human resources to foretell their occurrence and anticipate the aggressions. In this work, we propose to apply Machine Learning (ML) techniques to create models that accurately predict the recidivism risk o...

Reliability measures associated to machine learning model predictions are critical to reinforcing user confidence in artificial intelligence. Therefore, those models that are able to provide not only predictions, but also reliability enjoy greater popularity. In the field of recommender systems, reliability is crucial, since users tend to prefer th...

Context
The qualitative research on empirical software engineering that uses Grounded Theory is increasing (GT). The trustworthiness, rigor, and transparency of GT qualitative data analysis can benefit, among others, when multiple analysts juxtapose diverse perspectives and collaborate to develop a common code frame based on a consensual and consis...

Generative Adversarial Networks (GANs) are powerful Machine Learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable and typically it is necessary to implement several accessory heuristics to the networks to reach an acceptab...

Harmful algal blooms (HABs) are a growing concern to public health and aquatic ecosystems. Long-term water monitoring conducted by hand poses several limitations to the proper implementation of water safety plans. This work combines automatic high-frequency monitoring (AFHM) systems with machine learning (ML) techniques to build a data-driven chlor...

IoT edge computing is a new computing paradigm “in the IoT domain” for performing calculations and processing at the edge of the network, closer to the user and the source of the data. This paradigm is relatively recent, and, together with cloud and fog computing, there may be some confusion about its meaning and implications. This paper aims to he...

We describe the geometry of the character variety of representations of the knot group $\Gamma_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation...

In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into SL4(k), for any algebraically closed field k of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any...

Solving the convergence issues of Generative Adversarial Networks (GANs) is one of the most outstanding problems in generative models. In this work, we propose a novel activation function to be used as output of the generator agent. This activation function is based on the Smirnov probabilistic transformation and it is specifically designed to impr...

With the latest advances in deep learning 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 depend heavily on the breadth and consistency of the datasets used in training. These types of characteristic are not usually abundant in...

In this paper we investigate the problem of constructing Topological Quantum Field Theories (TQFTs) to quantize algebraic invariants. We provide necessary conditions for quantizability based on Euler characteristics and, in the case of surfaces, also sufficient conditions in terms of almost-TQFTs and almost-Frobenius algebras. As an application, we...

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 pro...

Due to the growing rise of cyber attacks in the Internet, the demand of accurate intrusion detection systems (IDS) to prevent these vulnerabilities is increasing. To this aim, Machine Learning (ML) components have been proposed as an efficient and effective solution. However, its applicability scope is limited by two important issues: (i) the short...

In the context of recommender systems based on collaborative filtering (CF), obtaining accurate neighborhoods of the items of the datasets is relevant. Beyond particular individual recommendations, knowing these neighbors is fundamental for adding differentiating factors to recommendations, such as explainability, detecting shilling attacks, visual...

Traditionally, recommender systems have been approached as regression models aiming to predict the score that a user would give to a particular item. In this work, we propose a recommender system that tackles the problem as a classification task instead of as a regression. The new model, Dirichlet Matrix Factorization (DirMF), provides not only a p...

In this paper, we extend the Topological Quantum Field Theory developed by Gonz\'alez-Prieto, Logares and Mu\~noz for computing virtual classes of 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 s...

Solving the convergence issues of Generative Adversarial Networks (GANs) is one of the most outstanding problems in generative models. In this work, we propose a novel activation function to be used as output of the generator agent. This activation function is based on the Smirnov probabilistic transformation and it is specifically designed to impr...

italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Context
: DevOps can be defined as a cultural movement to improve and accelerate the delivery of business value by making the collaboration between development and operations effective.
Objective
: This paper aims to help practitioners and researcher...

Due to the growing rise of cyber attacks in the Internet, flow-based data sets are crucial to increase the performance of the Machine Learning (ML) components that run in network-based intrusion detection systems (IDS). To overcome the existing network traffic data shortage in attack analysis, recent works propose Generative Adversarial Networks (G...

Deep learning provides accurate collaborative filtering models to improve recommender system results. Deep matrix factorization and their related collaborative neural networks are the state-of-art in the field; nevertheless, both models lack the necessary stochasticity to create the robust, continuous, and structured latent spaces that variational...

In recent years, the qualitative research on empirical software engineering that applies Grounded Theory is increasing. Grounded Theory (GT) is a technique for developing theory inductively e iteratively from qualitative data based on theoretical sampling, coding, constant comparison, memoing, and saturation, as main characteristics. Large or contr...

DevOps is becoming a main competency required by the software industry. However, academic institutions have been slow to provide DevOps training in software engineering (SE) curricula. One reason for this is the fact that the problems addressed by DevOps may be hard to understand to students who have not previously worked in the industry or on proj...

Gender-based crime is one of the most concerning scourges of contemporary society. Governments worldwide have invested lots of economic and human resources to radically eliminate this threat. Despite these efforts, providing accurate predictions of the risk that a victim of gender violence has of being attacked again is still a very hard open probl...

Extracting demographic features from hidden factors is an innovative concept that provides multiple and relevant applications. The matrix factorization model generates factors which do not incorporate semantic knowledge. Extracting the existing nonlinear relations between hidden factors and demographic information is a challenging task that can not...

We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of...

The lack of bias management in Recommender Systems leads to minority groups receiving unfair recommendations. Moreover, the trade-off between equity and precision makes it difficult to obtain recommendations that meet both criteria. Here we propose a Deep Learning based Collaborative Filtering algorithm that provides recommendations with an optimum...

We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C].

Context
DevOps can be defined as a cultural movement to improve and accelerate the delivery of business value by making the collaboration between development and operations effective. Although this movement is relatively recent, there exist an intensive research around DevOps. However, the real reasons why companies move to DevOps and the results t...

Generative Adversarial Networks (GANs) are powerful machine learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable, and typically, it is necessary to implement several accessory heuristics to the networks to reach acceptabl...

Context: DevOps can be defined as a cultural movement to improve and accelerate the delivery of business value by making the collaboration between development and operations effective. Objective: This paper aims to help practitioners and researchers to better understand the organizational structure and characteristics of teams adopting DevOps. Meth...

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 thei...

The aim of this chapter is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points...

Beyond accuracy, quality measures are gaining importance in modern recommender systems, with reliability being one of the most important indicators in the context of collaborative filtering. This paper proposes Bernoulli Matrix Factorization (BeMF), which is a matrix factorization model, to provide both prediction values and reliability values. BeM...

In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the pa...

Recommender Systems (RSs) have become an essential tool for the information society. Their incorporation into everyday life has allowed service providers to alleviate the information overload problem to which citizens are exposed. Every minute, hundreds of hours of video are posted on YouTube, thousands of products are purchased on Amazon, tens of...

Internet of Things (IoT) projects are increasing in size over time, and some of them are growing to reach the whole world. Sensor arrays are deployed world-wide and their data is sent to the cloud, making use of the Internet. These huge networks can be used to improve the quality of life of the humanity by continuously monitoring many useful indica...

In recent years, the research on empirical software engineering that uses qualitative data analysis (e.g. thematic analysis, content analysis, and grounded theory) is increasing. However, most of this research does not deep into the reliability and validity of findings, specifically in the reliability of coding, despite there exist a variety of sta...

Providing useful information to the users by recommending highly demanded products and services is a fundamental part of the business of many top tier companies. Recommender Systems make use of many sources of information to provide users with accurate predictions and novel recommendations of items. Here we propose, DeepMF, a novel collaborative fi...

In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of SL2(C)-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generating a recursive patter...

Extracting demographic features from hidden factors is an innovative concept that provides multiple and relevant applications. The matrix factorization model generates factors which do not incorporate semantic knowledge. This paper provides a deep learning-based method: DeepUnHide, able to extract demographic information from the users and items fa...

The lack of bias management in Recommender Systems leads to minority groups receiving unfair recommendations. Moreover, the trade-off between equity and precision makes it difficult to obtain recommendations that meet both criteria. Here we propose a Deep Learning based Collaborative Filtering algorithm that provides recommendations with an optimum...

Recommender Systems are giving increasing importance to the beyond accuracy quality measures, and reliability is one of the most important in the Collaborative Filtering context. This paper proposes Bernoulli Matrix Factorization (BeMF), a matrix factorization model to provide both prediction values and reliability ones. This is a very innovative a...

In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into SL_4(k), for any algebraically closed field k. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This...

DevOps can be defined as a cultural movement and a technical solution to improve and accelerate the delivery of business value by making the collaboration between development and operations effective, which is rapidly spreading in software industry. However this movement is relatively recent, being necessary more empirical evidence about the real r...

The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points ov...

Recommender systems aim to estimate the judgment or opinion that a user might offer to an item. Matrix-factorization-based collaborative filtering typifies both users and items as vectors of factors inferred from item rating patterns. This method finds latent structure in the data, assuming that observations lie close to a low-dimensional latent sp...

In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generati...

This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro-geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we dev...

In this paper, we use lax monoidal TQFTs as an effective computational method for motivic classes of representation varieties. In particular, we perform the calculation for parabolic $\mathrm{SL}_2(\mathbb{C})$-representation varieties over a closed orientable surface of arbitrary genus and any number of marked points with holonomies of Jordan type...

In this paper, we study a weaker version of algebraic quotient for the action of an algebraic group on an algebraic variety that is well behaved under stratification. Focusing on the topological properties of these quotients, we obtain a series of results about their structure and uniqueness. As an application, we compute the Deligne-Hodge polynomi...

We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group G. We also extend the result to the parabolic case with any number of punctures and arbitrary monodromies. As a byproduct, we obtain formula...

In this work, we shall study a special kind of algebraic varieties, the character varieties, and we will compute an algebro-geometric invariant of this varieties, known as the Deligne-Hodge polynomial or E-polynomial. This character varieties arise as moduli spaces of representations of the fundamental group of a compact Riemann surface with some r...