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Structure and Requirements of the Master’s Program

List of Courses (6 ECTS each) for year 25/26

First Semester

Machine Learning: Models and Applications

Machine learning combines statistics, optimization, and computing to recognize complex patterns in data. It enables the development of systems that automatically improve their performance, revolutionizing industries and scientific research.

Language: English

Specific Contents

  • Introduction and mathematical foundations: machine learning and statistical learning. Iterative methods for function optimization. Classical statistical inference.
  • Types of statistical learning problems: supervised (regression and classification), unsupervised (approximate density estimation), reinforcement, and hybrid scenarios (semi-supervised).
  • Models for supervised learning: neural networks (multilayer perceptron and deep architectures), support vector machines, decision trees, random forests, Gaussian processes, k-nearest neighbors, etc.
  • Models for unsupervised learning: dimensionality reduction (PCA, SVD, DMD), clustering models (k-means, hierarchical, self-organizing maps).
  • Applications in engineering and social sciences.

Professor: TBA

Ordinary Differential Equations and Applications

Ordinary differential equations are essential for describing the evolution of physical or biological systems over time. Their study allows for understanding, predicting, and controlling dynamic phenomena in engineering, science, and technology.

Language: English

Specific Contents

  • Non-autonomous linear systems. Periodic systems and Floquet theory.
  • Nonlinear systems: qualitative theory, invariant manifolds, topological equivalence.
  • Bifurcation theory.
  • Singular perturbations.
  • Chaotic phenomena.
  • Applications in science and engineering: mechanics and control of nonlinear systems, population dynamics.

Professor: TBA

Partial Differential Equations and Their Numerical Approximation

Partial differential equations are used to model phenomena involving multiple spatial and temporal variables. Learning to solve them numerically is key in fields such as engineering, computational physics, and computer simulation.

Language: Spanish

Specific Contents

  • Modeling the main problems that give rise to PDEs: conservation laws, diffusion equations, convection-diffusion-reaction equations.
  • Weak solutions for linear models. Variational methods for existence and uniqueness.
  • Variational approximation methods: Galerkin.
  • Approximation of elliptic equations: finite differences, collocation, spectral methods, and finite elements.
  • Approximation of parabolic problems. Explicit and implicit time methods.
  • Approximation of hyperbolic problems.

Professor: TBA

Advanced Statistics

Advanced statistics provides robust methods for analyzing complex data and extracting reliable conclusions. Its mastery is vital in areas such as data science, scientific research, and business decision-making.

Language: Spanish

Specific Contents

  • Stochastic modeling.
  • Computational and visualization tools. Data mining.
  • Statistical programming.
  • Time series. ARIMA models.
  • Exponential smoothing techniques.
  • Monte Carlo methods.
  • High-dimensional problems.
  • Stochastic optimization algorithms.

Professor: TBA

Advanced Modeling

This course delves into mathematical models for describing complex phenomena in fluid mechanics, continuous structures, and their dynamic behavior. It is essential for researchers and professionals in engineering and physics.

Language: Spanish

Specific Contents

  • Continuum mechanics. General conservation laws.
  • Conservation laws for Newtonian fluids. Main models of fluid dynamics. Non-dimensionalization.
  • Incompressible ideal flows. Irrotational and potential flows.
  • Incompressible viscous flows. Navier-Stokes equations.
  • Turbulent flows. Closure problem and introduction to main turbulence models.
  • Equations for beams and stationary plates.
  • Dynamic models. Stability analysis.

Professor: TBA

Advanced Algebra

This course delves into ring theory, modules, and modern techniques in cryptography and solving polynomial systems. Its study is key for the development of abstract mathematics and its computational applications.

Language: English

Specific Contents

  • Review of rings, modules, localization, and spectrum.
  • Noetherian rings and modules.
  • Integral dependence, finite morphisms, the going-up theorem, and Noether’s lemma.
  • Discrete valuation rings and Dedekind domains. Finiteness theorem. Resolution of singularities.
  • Completion, Artin-Rees lemma.
  • Cryptography and computational ideal theory.
  • Solving multivariate polynomial systems.

Professor: TBA

Advanced Analysis

This course addresses deep concepts of modern analysis, such as distributions and Sobolev spaces, providing the foundations for the study of mathematical problems in theoretical physics, differential equations, and integral transforms.

Language: English

Specific Contents

  • Distributions and Sobolev spaces.
  • Review of the Fourier transform in Euclidean space and in locally compact abelian groups.
  • Singular integrals and pseudodifferential operators.
  • Entire functions and the Paley-Wiener theorem. Uncertainty principles.
  • Hardy spaces and Nevanlinna theory.
  • Discrete Fourier analysis, sampling theory, wavelets, and applications to image and signal processing.

Professor: TBA

Advanced Differential and Complex Geometry

Differential and complex geometry provide tools to describe the deep structure of surfaces and manifolds, with applications in relativity, field theory, and other advanced areas of physics and pure mathematics.

Language: English

Specific Contents

  • Riemannian manifolds, geodesics, Hopf-Rinow theorem, hypersurfaces.
  • Topology and curvature, constant curvature manifolds, Hadamard and Chern-Gauss-Bonnet theorems.
  • Conformal structures.
  • Complex manifolds. Riemann surfaces.
  • Analytic and meromorphic functions on manifolds, uniformization theorem.
  • Divisors and the Riemann-Roch theorem.

Professor: TBA

Advanced Dynamical Systems

Dynamical systems study how a system evolves over time, from the emergence of chaos to the existence of attractors in finite and infinite-dimensional contexts. They are relevant in mechanics, meteorology, and complex modeling.

Language: English

Specific Contents

  • Review of basic concepts in discrete and continuous dynamical systems.
  • Invariant sets. Attractors.
  • Hamiltonian fields and applications.
  • Bifurcations: codimension. Chaos and chaotic phenomena.
  • Ergodicity, recurrence, and mixing. Entropy.
  • Infinite-dimensional dynamics. Semigroups of linear operators. Applications to evolution equations. Infinite-dimensional attractors.

Professor: TBA

Advanced Topology

Topology is concerned with studying the fundamental properties of spaces that are preserved under continuous deformation. This advanced course covers homology and cohomology, a basis for fields such as algebraic geometry and knot theory.

Language: Spanish

Specific Contents

  • Homology.
  • Cohomology.
  • Universal coefficient theorems, Künneth theorem, and duality.
  • Notable exact sequences: of the closed subspace, Mayer-Vietoris, and Gysin isomorphism.
  • Explicit calculations, fixed point theorem.

Professor: TBA

Second Semester

Advanced Extension of Statistics and Data Science

This course deepens techniques and statistical algorithms for handling and processing large volumes of data. Unsupervised and deep learning methods are applied in highly complex environments.

Language: English

Specific Contents

  • Specialized data science concepts to generate machine learning models considering probabilistic and statistical models.
  • Advanced algorithms focused on big data processing, autoencoders, and clustering techniques.
  • Advanced unsupervised learning algorithms and applications for massive data with many variables or complex dynamics.
  • Deep learning techniques for predictive, classification, and reconstruction models. Difference between classical and generative models.
  • Extension and generalization of deep learning models: models based on physical principles.
  • Reinforcement learning techniques. Industrial and engineering applications.

Professor: TBA

Advanced Extension of Algebra, Geometry, and Arithmetic

This course expands traditional concepts of algebra, geometry, and arithmetic, integrating computational methods and advanced techniques to address complex problems in mathematics and its applications.

Language: Spanish

Specific Contents

  • Sheaf cohomology and finiteness theorem.
  • Duality theory, dualizing sheaf, and the Riemann-Roch theorem.
  • Sheaf of differentials and calculation of the dualizing sheaf.
  • Projective, injective, and flat modules; torsion and extensions.
  • Koszul complexes, regular sequences, and Cohen-Macaulay rings.
  • Regular rings and Serre’s theorem.
  • Computational methods in noncommutative algebra: factorization and Ore extensions.
  • Gröbner bases in Poincaré-Birkhoff-Witt rings, quantum groups, differential and difference algebra, characteristic sets.

Professor: TBA

Mathematical Physics

This course explores the intersection between theoretical physics and modern mathematical tools, covering field theories, relativity, and quantum mechanics.

Language: Spanish

Specific Contents

  • Newtonian gravitation, spacetime, representation of matter, and geometric formulation.
  • Minkowski spacetime.
  • Electromagnetism.
  • General relativity, stress-energy tensor, and Einstein’s equation.
  • Introduction to quantum mechanics, observables.
  • Schrödinger equation, spin, and the hydrogen atom.

Professor: TBA

Note:

  • The final course offerings may vary each academic year.
  • Some subjects are taught in English. For more information click on the corresponding subject above.

According to the presented offering, students must complete:

  1. Either 18 credits provided by the 3 courses of the Applied Mathematics Block in the first semester:
    • Ordinary Differential Equations and Applications.
    • Partial Differential Equations and Their Numerical Approximation.
    • Advanced Statistics.
    or 18 credits provided by the 3 courses of the Fundamental Mathematics Block in the first semester:
    • Advanced Analysis.
    • Advanced Algebra.
    • Advanced Differential and Complex Geometry.
  2. Another 24 credits among the remaining courses, including those not selected above, and external academic internships (maximum 6 ECTS).
  3. Finally, 18 credits for the Master’s Thesis (TFM). As a guideline, consult GeM Bachelor Final Project proposals.