Research

Research Interest

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.

In addition, I also research in theoretical aspects of machine learning and big data. Particularly, I am interested in collaborative filtering-based recommender systems as well as applications of highly parallelizable deep learning methods and generative adversarial networks (GANs) to real-time problems. Finally, I study the transference of geometric techniques to dimensional reduction problems and manifold learning.

Papers

Algebraic Geometry

1. (with J. Martínez and V. Muñoz) Character varieties of torus links. Preprint arXiv:2402.12286.
2. (with M. Hablicsek and J. Vogel) Arithmetic-Geometric Correspondence of Character Stacks via Topological Quantum Field Theory. Preprint arXiv:2309.15331.
3. (with M. Logares, J. Martínez and V. Muñoz) Stratification of SU(r)-character varieties of twisted Hopf links.
   To appear in Contemporary Mathematics of the American Mathematical Society
4. (with J. Martínez and V. Muñoz) Geometry of SU(3)-character varieties of torus knots.
   Topology and its Applications, 108586.
5. Quantization of algebraic invariants through Topological Quantum Field Theories.
   Journal of Geometry and Physics, 189, 104849.
6. (with V. Muñoz) Representation varieties of twisted Hopf links.
   Mediterranean Journal of Mathematics, 20(2), 89.
7. (with M. Hablicsek and J. Vogel) Virtual classes of character stacks.
   Preprint arXiv:2201.08699.
8. (with V. Muñoz) The point counting problem in representation varieties of torus knots.
   Montes Taurus J. Pure Appl. Math. 4 (3), 114-130, 2022.
9. (with M. Logares and V. Muñoz) Motive of the representation varieties of torus knots for low rank affine groups.
   Analysis, Geometry, Nonlinear optimization and Applications (Th.M. Rassias, P. Pardalos eds.), World Scientific.
10. (with M. Logares) On character varieties of singular manifolds.
    Research in the Mathematical Sciences, 10.
11. (with V. Muñoz) Motive of the SL₄-character variety of torus knots.
    Journal of Algebra, 610, 852-895.
12. (with M. Logares and V. Muñoz) Representation variety for the rank one affine group.
    Mathematical Analysis in Interdisciplinary Research, 381-416, Springer, Cham, 2021.
13. Virtual classes of parabolic SL₂(C)-character varieties.
    Advances in Mathematics, 368, 107148.
14. Pseudo-quotients of algebraic actions and their application to character varieties.
    Communications in Contemporary Mathematics.
15. Motivic theory of representation varieties via Topological Quantum Field Theories.
    Preprint arXiv:1810.09714.
16. (with M. Logares and V. Muñoz) A lax monoidal Topological Quantum Field Theory for representation varieties.
    Bulletin des Sciences Mathématiques, Vol. 161, 2020, 102871.

Theoretical and Applied Machine Learning

1. (with R. Lara-Cabrera, D. Trujillo, F. Ortega and D. Pérez-López) Manifoldy: An evaluation framework for dimensionality reduction through sectional curvature.
   Preprint (2023).
2. (with J. Dueñas, F. Ortega and D. Pérez-López) Incorporating Recklessness to Collaborative Filtering based Recommender Systems.
   Preprint (2023).
3. (with A. Gutiérrez, R. Lara-Cabrera and F. Ortega) ResBeMF: Improving Prediction Coverage of Classification-based Collaborative Filtering.
   Preprint (2022).
4. (with A. Brú, J. C. Nuño and J. L. González-Álvarez) Hybrid machine learning methods for risk assessment in gender-based crime.
   Knowledge-Based Systems (2022).
5. (with A. Mozo, S. Gómez-Canaval and E. Talavera) Improving the quality of generative models through Smirnov transformation.
   Information Sciences 609.
6. (with A. Mozo, A. Pastor, S. Gómez-Canaval and E. Talavera) Synthetic flow-based cryptomining attack generation through Generative Adversarial Networks.
   Scientific Reports 12, 2091 (2022).
7. (with E. Talavera, G. Iglesias, A. Mozo and S. Gómez-Canaval) Data Augmentation techniques in time series domain: A survey and taxonomy.
   Neural Computing and Applications 35, 10123–10145.
8. (with Á. González, R. Lara-Cabrera and F. Ortega) Dirichlet Matrix Factorization: A Reliable Classification-based Recommender System.
   Appl. Sci. 2022, 12(3).
9. (with A. Mozo, J. Morón-López, S. Vakaruk et al.) Chlorophyll soft-sensor: machine learning models for algal bloom predictions.
   Sci. Rep. 12.1 (2022): 1-23.
10. (with J. Bobadilla, R. Gutiérrez and F. Ortega) Deep Variational Models for Collaborative Filtering-based Recommender Systems.
    Neural Computing and Applications.
11. (with J. Bobadilla, R. Lara-Cabrera and F. Ortega) Deep Learning Approach to Obtain Collaborative Filtering Neighborhoods.
    Neural Computing and Applications 34, 2939-2951 (2022).
12. (with A. Mozo, E. Talavera and S. Gómez-Canaval) Dynamics of Fourier Modes in Torus Generative Adversarial Networks.
    Math. 2021, 9, 325.
13. (with J. Bobadilla, R. Lara-Cabrera and F. Ortega) Providing reliability in Recommender Systems through Bernoulli Matrix Factorization.
    Information Sciences, 553 110-128.
14. (with J. Bobadilla, R. Lara-Cabrera and F. Ortega) Deep learning feature selection to unhide demographic recommender systems factors.
    Neural Computing and Applications, 1-18.
15. (with J. Bobadilla, R. Lara-Cabrera and F. Ortega) DeepFair: Deep Learning for Improving Fairness in Recommender Systems.
    International Journal of Interactive Multimedia and Artificial Intelligence.
16. (with J. Bobadilla, A. Gutiérrez and F. Ortega) Collaborative filtering to predict sensor array values in large IoT networks.
    Sensors 2020, 20(16), 4628.
17. (with R. Lara-Cabrera and F. Ortega) Deep Matrix Factorization approach for Collaborative Filtering Recommender Systems.
    Appl. Sci. 2020, 10(14), 4926.
18. (with J. Bobadilla, R. Lara-Cabrera and F. Ortega) Evolving Matrix Factorization based Collaborative Filtering using Genetic Programming.
    Appl. Sci. 2020, 10(2), 675.

Statistical Techniques in Software Engineering

1. (with J. Pérez, J. Díaz, J. Berrocal and R. López-Viana) Edge Computing — A Grounded Theory Study.
   Computing (2022)
2. (with J. Díaz, J. Perez and C. Gallardo) Applying Inter-rater Reliability and Agreement in Grounded Theory Studies in Software Engineering.
   J. Syst. Softw. 195 (2023) 111520.
3. (with D. López-Fernández, J. Diaz, J. García and J. Perez) DevOps Team Structures: Characterization and Implications.
   IEEE Transactions on Software Engineering.
4. (with J. Perez, D. López-Fernández, J. Diaz, J. García and A. Yagüe) DevOps Research-based Teaching Using QualitativeResearch and Inter-Coder Agreement.
   IEEE Transactions on Software Engineering.
5. (with J. Perez, J. Diaz and D. López-Fernández) Inter-Coder Agreement for Improving Reliability in Software Engineering Qualitative Research.
   J. Syst. Softw. 202 111707.
6. (with J. Diaz, D. López-Fernández and J. Perez) Why are many businesses instilling a DevOps culture into their organization?
   Empir. Softw. Eng. 2021, 26(25)

Books

• (with V. Muñoz and J. A. Rojo) Geometry and Topology of Manifolds: Surfaces and Beyond.
  Graduate Studies in Mathematics (American Mathematical Society). ISBN: 978-1-4704-6132-4

PhD Thesis

• Topological Quantum Field Theories for Character Varieties. Slides.

Master’s and Bachelor’s Thesis

• Master’s Thesis: Deligne-Hodge Polynomials of Character Varieties of Doubly Periodic Instantons. Slides. Published at Ediciones UAM. 2016.
• Bachelor’s Thesis: The h-cobordism theorem (in Spanish). Slides.

Lecture Notes

• Symplectic Geometry and Moment Maps with a View Towards Morse-Bott Theory.
• Derived Functors and Sheaf Cohomology (in Spanish).
• Logic for Artificial Intelligence (in Spanish).
• Detailed proof of the Snake’s Lemma (in Spanish).

Links to my profiles

• Google Scholar profile.
• ORCID profild.
• ReseachGate profile.
• dblp profile.