Escuela de Ingeniería, Ciencia y Tecnología

Computational science refers to the set of techniques, theories, and tools based on numerical analysis and theoretical computer science. It is used to solve mathematical models of complex systems in science and engineering, providing a mathematical and computational base for calculating and numerically solving equations. It particularly focuses on the areas of materials science, molecular simulation, the study of nanoscale systems, tensor networks, and computational biology. The tools used to address these problems include tensor networks, stochastic algorithms, and first principles methods, among others.

In many fields of science and technology, developing new knowledge depends on analyzing large amounts of data to extract information that supports the evaluation of research hypotheses, decision-making, and system design, among others. Data science provides mathematical, statistical, algorithmic, and technological tools that allow researchers to identify patterns, rigorously evaluate alternative hypotheses about a phenomenon, and compare multiple designs for a system. This area of research may be applied to areas as diverse as social, biological, medical, and economic sciences, as well as to the design of systems across multiple fields within engineering. Within this area, we have proposed methods for evaluating the sensitivity of hypothesis tests, as well as for modeling individuals’ consumption and its effects. We have also developed estimation models for correlated phenomena, with applications for modeling distributed computer systems and energy resources. Furthermore, we have proposed probabilistic models based on operational relationships for the design of software systems, as well as corresponding statistical methods that enable data parameterization. In addition to this, we work on applying Monte Carlo methods using Markov chains. 

Cyber security focuses on ensuring the security and reliability of both information systems and cyber-physical systems. In this research area, we study secret-key cryptography (where two agents share the same key), and public-key cryptography (where the encryption works similarly to having a public email and a secret password). We research post-quantum cryptography, aiming to discover new protocols based on mathematics that cannot be solved by quantum or classic computing. We also study the analysis of software design and implementation at various levels of abstraction. This is important as most attacks involve exploiting design flaws or implementation errors (for example, exploiting vulnerabilities in database queries through SQL injections). In this research area, we explore model verification techniques, programming languages, and automated software testing. We apply them to cyber-physical systems to improve the quality and security of software. 

Complex systems are characterized by the absence of reductionist approaches that enable them to be understood on a global level. Examples of such systems include human behavior in society, modern computer systems (physical and online), dynamic systems (physical, cybernetic, social, and economic), and many-body systems. Using knowledge from game theory and graphs, probabilistic techniques, statistics, and optimization, our goal is to model, design, and, in some cases, control complex systems in the different contexts discussed previously. In particular, we aim to: 

(i) Study and understand the behavior of human beings in society, which can lead to collective events that cannot be produced by a single individual. (ii) Develop mathematical models for designing, controlling, and optimizing complex computer systems, based on the technological platform used, and considering all layers of resources and their interactions. (iii) Research new strategies for addressing large-scale control problems, using decentralized schemes that use controller networks, multi-agent systems, and distributed optimization algorithms. (iv) Model and simulate many-body systems by developing optimization techniques that enable the manipulation of the order and dynamics of these physical systems. 

Machine learning, a branch of artificial intelligence, aims to create algorithms and methodologies using data, enabling computers to establish rules for clearly defined problems. Machine learning is generally developed by training mathematical models using real data and/or providing prior knowledge. In this area, we have developed computer systems that are capable of adapting to changing scenarios, where users and their requests are dynamic and unpredictable. The system is thus required to autonomously adapt to ensure the reliability and quality of its response. We have also focused on using techniques based on information theory to be able to systematically analyze the fundamental principles of machine learning. This will provide a global theoretical foundation that will serve to support the development of new interpretable computer architectures and models. On the other hand, the development of interpretable models has been approached from another perspective, where oblique projections are used to recover the interpretability of kernel-based models. These models have been used in medical applications for monitoring premature babies. 

In order to understand the teaching and learning processes in the field of mathematics, it is recommended to explore issues that are centered around reflecting on one’s own teaching methods, using technology in teaching and learning, curriculum design, innovation, and quality-oriented training. All of this means making research more accessible and relevant to our students’ everyday lives, respecting diversity, and adopting a naturalistic, context-sensitive approach. We are developing and/or improving methods for teaching various statistical topics to students studying university courses where mathematics is not the main focus. In this context, a method for teaching Bayes’ theorem was developed using a didactic model based on heuristic methods derived from contingency tables. Additionally, efforts are being made to improve the teaching of hypothesis testing, emphasizing its methodological role in quantitative research by incorporating the random nature of the P value and meta-analyses into the courses. 

Mathematical analysis is a very extensive branch of mathematics that studies real numbers, complex numbers, and constructions derived from them, as well as the properties of functions connecting these sets. 

Topology is based on the study of the properties of geometric bodies that remain unchanged by continuous transformations. Topology is concerned with concepts such as proximity, the counting of holes, the type of consistency that an object exhibits, and comparing and classifying objects. Among its multiple attributes, notable aspects include connectivity, compactness, metricity, and metrizability, etc. 

This research area aims to formally develop the mathematical foundations necessary for applying mathematical analysis and topology. It specifically focuses on studying the properties of various linear operators, such as compactness, boundaries, stability under additive perturbations, generalizations of topological structures, and the behavior of topological properties in weak structures.