Structure and Requirements of the Master’s Program<\/title>\n <style>\n \/* Styles for tracks *\/\n .itinerario-aplicada {\n background-color: #f9f3e7; \/* Brownish-yellow tone *\/\n border: 1px solid #ccc;\n border-radius: 6px;\n }\n .itinerario-fundamental {\n background-color: #e7f3f9; \/* Light blue tone *\/\n border: 1px solid #ccc;\n padding: 1em;\n margin: 0.5em 0;\n border-radius: 6px;\n }\n \n \/* General styles *\/\n body {\n font-family: sans-serif;\n line-height: 1.4;\n }\n h2 {\n margin-top: 0;\n }\n .info-container {\n border: 1px solid #ddd;\n padding: 1em;\n border-radius: 6px;\n background-color: #fafafa;\n margin-bottom: 1.5em;\n }\n \/* Background colors as defined *\/\n details[data-itinerario=\"I\"] {\n background-color: #f9f3e7;\n }\n details[data-itinerario=\"II\"] {\n background-color: #e7f3f9;\n }\n details[data-itinerario=\"III\"] {\n background-color: #d9f6ff;\n }\n details[data-itinerario=\"IV\"] {\n background-color: #d9fff7;\n }\n details summary {\n 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{\n border: 3px solid #7aaee0;\n background-color: #dceef8;\n }\n .obligatoria-label {\n font-size: 0.8em;\n font-weight: bold;\n padding: 0.2em 0.5em;\n border-radius: 4px;\n margin-left: 0.5em;\n }\n details.obligatoria[data-itinerario-highlight=\"aplicada\"] .obligatoria-label {\n background-color: #a67c52;\n color: #fff;\n }\n details.obligatoria[data-itinerario-highlight=\"fundamental\"] .obligatoria-label {\n background-color: #7aaee0;\n color: #fff;\n }\n \/* Container in two adjacent columns *\/\n .container {\n display: flex;\n flex-wrap: wrap;\n gap: 2em;\n }\n .column {\n flex: 1;\n min-width: 300px;\n }\n \/* Button container: center the buttons next to each other *\/\n .button-container {\n text-align: center;\n margin-bottom: 1em;\n }\n \/* Big buttons for highlighting *\/\n .toggle-btn {\n display: inline-block;\n padding: 1em 2em;\n margin: 0.5em;\n font-size: 1.2em;\n font-weight: bold;\n border: none;\n border-radius: 6px;\n cursor: pointer;\n }\n .toggle-btn.applied {\n background-color: #d6c2af;\n color: #000;\n }\n .toggle-btn.fundamental {\n background-color: #c4e0f5;\n color: #000;\n }\n \/* Highlight styles: change border only *\/\n details[data-itinerario-highlight=\"aplicada\"].highlight {\n border: 3px solid #a67c52;\n }\n details[data-itinerario-highlight=\"fundamental\"].highlight {\n border: 3px solid #7aaee0;\n }\n <\/style>\n\n <script>\n \/\/ Get references to the buttons\n const btnAplicada = document.getElementById(\"btnAplicada\");\n const btnFundamental = document.getElementById(\"btnFundamental\");\n\n \/\/ Function to remove highlight from all details\n function removeHighlight() {\n document.querySelectorAll(\"details\").forEach(detail => {\n detail.classList.remove(\"highlight\");\n });\n }\n\n \/\/ When the Applied Mathematics track button is clicked, add highlight only to details with data-itinerario=\"aplicada\"\n btnAplicada.addEventListener(\"click\", () => {\n removeHighlight();\n document.querySelectorAll('details[data-itinerario=\"aplicada\"]').forEach(detail => {\n detail.classList.add(\"highlight\");\n });\n });\n\n \/\/ When the Fundamental Mathematics track button is clicked, add highlight only to details with data-itinerario=\"fundamental\"\n btnFundamental.addEventListener(\"click\", () => {\n removeHighlight();\n document.querySelectorAll('details[data-itinerario=\"fundamental\"]').forEach(detail => {\n detail.classList.add(\"highlight\"); });\n });\n <\/script>\n\n <h2>List of Courses (6 ECTS each) for year 25\/26<\/h2>\n <div class=\"button-container\">\n <button id=\"btnAplicada\" class=\"toggle-btn applied\">Applied Mathematics Block<\/button>\n <button id=\"btnFundamental\" class=\"toggle-btn fundamental\">Fundamental Mathematics Block<\/button>\n <\/div>\n <div class=\"container\">\n <div class=\"column\">\n <h2>First Semester<\/h2>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\">\n <summary>Machine Learning: Models and Applications<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Introduction and mathematical foundations: machine learning and statistical learning. Iterative methods for function optimization. Classical statistical inference.<\/li>\n <li>Types of statistical learning problems: supervised (regression and classification), unsupervised (approximate density estimation), reinforcement, and hybrid scenarios (semi-supervised).<\/li>\n <li>Models for supervised learning: neural networks (multilayer perceptron and deep architectures), support vector machines, decision trees, random forests, Gaussian processes, k-nearest neighbors, etc.<\/li>\n <li>Models for unsupervised learning: dimensionality reduction (PCA, SVD, DMD), clustering models (k-means, hierarchical, self-organizing maps).<\/li>\n <li>Applications in engineering and social sciences.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000088_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\" class=\"obligatoria\">\n <summary>Ordinary Differential Equations and Applications<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Non-autonomous linear systems. Periodic systems and Floquet theory.<\/li>\n <li>Nonlinear systems: qualitative theory, invariant manifolds, topological equivalence.<\/li>\n <li>Bifurcation theory.<\/li>\n <li>Singular perturbations.<\/li>\n <li>Chaotic phenomena.<\/li>\n <li>Applications in science and engineering: mechanics and control of nonlinear systems, population dynamics.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000084_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\" class=\"obligatoria\">\n <summary>Partial Differential Equations and Their Numerical Approximation<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Modeling the main problems that give rise to PDEs: conservation laws, diffusion equations, convection-diffusion-reaction equations.<\/li>\n <li>Weak solutions for linear models. Variational methods for existence and uniqueness.<\/li>\n <li>Variational approximation methods: Galerkin.<\/li>\n <li>Approximation of elliptic equations: finite differences, collocation, spectral methods, and finite elements.<\/li>\n <li>Approximation of parabolic problems. Explicit and implicit time methods.<\/li>\n <li>Approximation of hyperbolic problems.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000084_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\" class=\"obligatoria\">\n <summary>Advanced Statistics<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Stochastic modeling.<\/li>\n <li>Computational and visualization tools. Data mining.<\/li>\n <li>Statistical programming.<\/li>\n <li>Time series. ARIMA models.<\/li>\n <li>Exponential smoothing techniques.<\/li>\n <li>Monte Carlo methods.<\/li>\n <li>High-dimensional problems.<\/li>\n <li>Stochastic optimization algorithms.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000085_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\">\n <summary>Advanced Modeling<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Continuum mechanics. General conservation laws.<\/li>\n <li>Conservation laws for Newtonian fluids. Main models of fluid dynamics. Non-dimensionalization.<\/li>\n <li>Incompressible ideal flows. Irrotational and potential flows.<\/li>\n <li>Incompressible viscous flows. Navier-Stokes equations.<\/li>\n <li>Turbulent flows. Closure problem and introduction to main turbulence models.<\/li>\n <li>Equations for beams and stationary plates.<\/li>\n <li>Dynamic models. Stability analysis.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000087_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\" class=\"obligatoria\">\n <summary>Advanced Algebra<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Review of rings, modules, localization, and spectrum.<\/li>\n <li>Noetherian rings and modules.<\/li>\n <li>Integral dependence, finite morphisms, the going-up theorem, and Noether’s lemma.<\/li>\n <li>Discrete valuation rings and Dedekind domains. Finiteness theorem. Resolution of singularities.<\/li>\n <li>Completion, Artin-Rees lemma.<\/li>\n <li>Cryptography and computational ideal theory.<\/li>\n <li>Solving multivariate polynomial systems.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000093_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\" class=\"obligatoria\">\n <summary>Advanced Analysis<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Distributions and Sobolev spaces.<\/li>\n <li>Review of the Fourier transform in Euclidean space and in locally compact abelian groups.<\/li>\n <li>Singular integrals and pseudodifferential operators.<\/li>\n <li>Entire functions and the Paley-Wiener theorem. Uncertainty principles.<\/li>\n <li>Hardy spaces and Nevanlinna theory.<\/li>\n <li>Discrete Fourier analysis, sampling theory, wavelets, and applications to image and signal processing.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000092_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\" class=\"obligatoria\">\n <summary>Advanced Differential and Complex Geometry<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Riemannian manifolds, geodesics, Hopf-Rinow theorem, hypersurfaces.<\/li>\n <li>Topology and curvature, constant curvature manifolds, Hadamard and Chern-Gauss-Bonnet theorems.<\/li>\n <li>Conformal structures.<\/li>\n <li>Complex manifolds. Riemann surfaces.<\/li>\n <li>Analytic and meromorphic functions on manifolds, uniformization theorem.<\/li>\n <li>Divisors and the Riemann-Roch theorem.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000094_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Advanced Dynamical Systems<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Review of basic concepts in discrete and continuous dynamical systems.<\/li>\n <li>Invariant sets. Attractors.<\/li>\n <li>Hamiltonian fields and applications.<\/li>\n <li>Bifurcations: codimension. Chaos and chaotic phenomena.<\/li>\n <li>Ergodicity, recurrence, and mixing. Entropy.<\/li>\n <li>Infinite-dimensional dynamics. Semigroups of linear operators. Applications to evolution equations. Infinite-dimensional attractors.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000097_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Advanced Topology<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Homology.<\/li>\n <li>Cohomology.<\/li>\n <li>Universal coefficient theorems, K\u00fcnneth theorem, and duality.<\/li>\n <li>Notable exact sequences: of the closed subspace, Mayer-Vietoris, and Gysin isomorphism.<\/li>\n <li>Explicit calculations, fixed point theorem.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000096_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <\/div>\n\n <div class=\"column\">\n <h2>Second Semester<\/h2>\n\n <details data-semester=\"II\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\">\n <summary>Advanced Extension of Statistics and Data Science<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Specialized data science concepts to generate machine learning models considering probabilistic and statistical models.<\/li>\n <li>Advanced algorithms focused on big data processing, autoencoders, and clustering techniques.<\/li>\n <li>Advanced unsupervised learning algorithms and applications for massive data with many variables or complex dynamics.<\/li>\n <li>Deep learning techniques for predictive, classification, and reconstruction models. Difference between classical and generative models.<\/li>\n <li>Extension and generalization of deep learning models: models based on physical principles.<\/li>\n <li>Reinforcement learning techniques. Industrial and engineering applications.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000091_2S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"II\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Advanced Extension of Algebra, Geometry, and Arithmetic<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Sheaf cohomology and finiteness theorem.<\/li>\n <li>Duality theory, dualizing sheaf, and the Riemann-Roch theorem.<\/li>\n <li>Sheaf of differentials and calculation of the dualizing sheaf.<\/li>\n <li>Projective, injective, and flat modules; torsion and extensions.<\/li>\n <li>Koszul complexes, regular sequences, and Cohen-Macaulay rings.<\/li>\n <li>Regular rings and Serre’s theorem.<\/li>\n <li>Computational methods in noncommutative algebra: factorization and Ore extensions.<\/li>\n <li>Gr\u00f6bner bases in Poincar\u00e9-Birkhoff-Witt rings, quantum groups, differential and difference algebra, characteristic sets.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000100_2S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"II\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Mathematical Physics<\/summary>\n <p>\n This course explores the intersection between theoretical physics and modern mathematical tools, covering field theories, relativity, and quantum mechanics.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Newtonian gravitation, spacetime, representation of matter, and geometric formulation.<\/li>\n <li>Minkowski spacetime.<\/li>\n <li>Electromagnetism.<\/li>\n <li>General relativity, stress-energy tensor, and Einstein’s equation.<\/li>\n <li>Introduction to quantum mechanics, observables.<\/li>\n <li>Schr\u00f6dinger equation, spin, and the hydrogen atom.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000099_2S_2025-26\">download<\/a><\/p>\n <\/details>\n <\/div>\n <\/div>\n\n <div class=\"info-container\">\n <p><strong>Note:<\/strong><\/p>\n <ul>\n <li>The final course offerings may vary each academic year.<\/li>\n <li>Some subjects are taught in English. For more information click on the corresponding subject above.<\/li>\n <\/ul>\n <p>According to the presented offering, students must complete:<\/p>\n <ol>\n <li>\n Either <strong>18 credits<\/strong> provided by the 3 courses of the <em>Applied Mathematics Block<\/em> in the first semester:\n <ul>\n <li>Ordinary Differential Equations and Applications.<\/li>\n <li>Partial Differential Equations and Their Numerical Approximation.<\/li>\n <li>Advanced Statistics.<\/li>\n <\/ul>\n or <strong>18 credits<\/strong> provided by the 3 courses of the <em>Fundamental Mathematics Block<\/em> in the first semester:\n <ul>\n <li>Advanced Analysis.<\/li>\n <li>Advanced Algebra.<\/li>\n <li>Advanced Differential and Complex Geometry.<\/li>\n <\/ul>\n <\/li>\n <li>Another <strong>24 credits<\/strong> among the remaining courses, including those not selected above, and external academic internships (maximum 6 ECTS).<\/li>\n <li>Finally, <strong>18 credits<\/strong> for the Master’s Thesis (TFM). <a href=\"https:\/\/matematicas.epes.upm.es\/master\/propuestas-de-tfm\/\" target=\"_blank\">Here<\/a> you may consult the topics proposed this year.<\/li>\n <\/ol>\n<ul>\n <li><a href=\"https:\/\/matematicas.epes.upm.es\/master\/wp-content\/uploads\/2025\/09\/Borrador_memoria_Def_y_anexo.pdf\" target=\"_blank\">Here<\/a> you can consult master’s description document (in spanish).<\/li>\n <\/ul>\n <\/div>\n\n <script>\n \/\/ Get references to the buttons\n const btnAplicada2 = document.getElementById(\"btnAplicada\");\n const btnFundamental2 = document.getElementById(\"btnFundamental\");\n\n \/\/ Function to remove highlight from all details\n function removeHighlight2() {\n document.querySelectorAll(\"details\").forEach(detail => {\n detail.classList.remove(\"highlight\");\n });\n }\n\n \/\/ Applied track button\n btnAplicada2.addEventListener(\"click\", () => {\n removeHighlight2();\n document.querySelectorAll('details[data-itinerario=\"aplicada\"]').forEach(detail => {\n detail.classList.add(\"highlight\");\n });\n });\n\n \/\/ Fundamental track button\n btnFundamental2.addEventListener(\"click\", () => {\n removeHighlight2();\n document.querySelectorAll('details[data-itinerario=\"fundamental\"]').forEach(detail => {\n detail.classList.add(\"highlight\");\n });\n });\n <\/script>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Structure and Requirements of the Master’s Program List of Courses (6 ECTS each) for year 25\/26 Applied Mathematics Block Fundamental Mathematics Block 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 … <a href=\"https:\/\/matematicas.epes.upm.es\/master\/en\/structure\/\">Continue reading <span class=\"meta-nav\">→<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-full-width.php","meta":{"footnotes":""},"class_list":["post-113","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages\/113","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/comments?post=113"}],"version-history":[{"count":21,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages\/113\/revisions"}],"predecessor-version":[{"id":586,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages\/113\/revisions\/586"}],"wp:attachment":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/media?parent=113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

{"id":113,"date":"2025-03-07T10:51:12","date_gmt":"2025-03-07T10:51:12","guid":{"rendered":"https:\/\/matematicas.epes.upm.es\/master\/?page_id=113"},"modified":"2025-10-08T18:24:51","modified_gmt":"2025-10-08T18:24:51","slug":"structure","status":"publish","type":"page","link":"https:\/\/matematicas.epes.upm.es\/master\/en\/structure\/","title":{"rendered":"Structure"},"content":{"rendered":"\n\n\n\n \n Structure and Requirements of the Master’s Program<\/title>\n <style>\n \/* Styles for tracks *\/\n .itinerario-aplicada {\n background-color: #f9f3e7; \/* Brownish-yellow tone *\/\n border: 1px solid #ccc;\n border-radius: 6px;\n }\n .itinerario-fundamental {\n background-color: #e7f3f9; \/* Light blue tone *\/\n border: 1px solid #ccc;\n padding: 1em;\n margin: 0.5em 0;\n border-radius: 6px;\n }\n \n \/* General styles *\/\n body {\n font-family: sans-serif;\n line-height: 1.4;\n }\n h2 {\n margin-top: 0;\n }\n .info-container {\n border: 1px solid #ddd;\n padding: 1em;\n 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{\n flex: 1;\n min-width: 300px;\n }\n \/* Button container: center the buttons next to each other *\/\n .button-container {\n text-align: center;\n margin-bottom: 1em;\n }\n \/* Big buttons for highlighting *\/\n .toggle-btn {\n display: inline-block;\n padding: 1em 2em;\n margin: 0.5em;\n font-size: 1.2em;\n font-weight: bold;\n border: none;\n border-radius: 6px;\n cursor: pointer;\n }\n .toggle-btn.applied {\n background-color: #d6c2af;\n color: #000;\n }\n .toggle-btn.fundamental {\n background-color: #c4e0f5;\n color: #000;\n }\n \/* Highlight styles: change border only *\/\n details[data-itinerario-highlight=\"aplicada\"].highlight {\n border: 3px solid #a67c52;\n }\n details[data-itinerario-highlight=\"fundamental\"].highlight {\n border: 3px solid #7aaee0;\n }\n <\/style>\n\n <script>\n \/\/ Get references to the buttons\n const btnAplicada = document.getElementById(\"btnAplicada\");\n const btnFundamental = document.getElementById(\"btnFundamental\");\n\n \/\/ Function to remove highlight from all details\n function removeHighlight() {\n document.querySelectorAll(\"details\").forEach(detail => {\n detail.classList.remove(\"highlight\");\n });\n }\n\n \/\/ When the Applied Mathematics track button is clicked, add highlight only to details with data-itinerario=\"aplicada\"\n btnAplicada.addEventListener(\"click\", () => {\n removeHighlight();\n document.querySelectorAll('details[data-itinerario=\"aplicada\"]').forEach(detail => {\n detail.classList.add(\"highlight\");\n });\n });\n\n \/\/ When the Fundamental Mathematics track button is clicked, add highlight only to details with data-itinerario=\"fundamental\"\n btnFundamental.addEventListener(\"click\", () => {\n removeHighlight();\n document.querySelectorAll('details[data-itinerario=\"fundamental\"]').forEach(detail => {\n detail.classList.add(\"highlight\"); });\n });\n <\/script>\n\n <h2>List of Courses (6 ECTS each) for year 25\/26<\/h2>\n <div class=\"button-container\">\n <button id=\"btnAplicada\" class=\"toggle-btn applied\">Applied Mathematics Block<\/button>\n <button id=\"btnFundamental\" class=\"toggle-btn fundamental\">Fundamental Mathematics Block<\/button>\n <\/div>\n <div class=\"container\">\n <div class=\"column\">\n <h2>First Semester<\/h2>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\">\n <summary>Machine Learning: Models and Applications<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Introduction and mathematical foundations: machine learning and statistical learning. Iterative methods for function optimization. Classical statistical inference.<\/li>\n <li>Types of statistical learning problems: supervised (regression and classification), unsupervised (approximate density estimation), reinforcement, and hybrid scenarios (semi-supervised).<\/li>\n <li>Models for supervised learning: neural networks (multilayer perceptron and deep architectures), support vector machines, decision trees, random forests, Gaussian processes, k-nearest neighbors, etc.<\/li>\n <li>Models for unsupervised learning: dimensionality reduction (PCA, SVD, DMD), clustering models (k-means, hierarchical, self-organizing maps).<\/li>\n <li>Applications in engineering and social sciences.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000088_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\" class=\"obligatoria\">\n <summary>Ordinary Differential Equations and Applications<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Non-autonomous linear systems. Periodic systems and Floquet theory.<\/li>\n <li>Nonlinear systems: qualitative theory, invariant manifolds, topological equivalence.<\/li>\n <li>Bifurcation theory.<\/li>\n <li>Singular perturbations.<\/li>\n <li>Chaotic phenomena.<\/li>\n <li>Applications in science and engineering: mechanics and control of nonlinear systems, population dynamics.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000084_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\" class=\"obligatoria\">\n <summary>Partial Differential Equations and Their Numerical Approximation<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Modeling the main problems that give rise to PDEs: conservation laws, diffusion equations, convection-diffusion-reaction equations.<\/li>\n <li>Weak solutions for linear models. Variational methods for existence and uniqueness.<\/li>\n <li>Variational approximation methods: Galerkin.<\/li>\n <li>Approximation of elliptic equations: finite differences, collocation, spectral methods, and finite elements.<\/li>\n <li>Approximation of parabolic problems. Explicit and implicit time methods.<\/li>\n <li>Approximation of hyperbolic problems.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000084_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\" class=\"obligatoria\">\n <summary>Advanced Statistics<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Stochastic modeling.<\/li>\n <li>Computational and visualization tools. Data mining.<\/li>\n <li>Statistical programming.<\/li>\n <li>Time series. ARIMA models.<\/li>\n <li>Exponential smoothing techniques.<\/li>\n <li>Monte Carlo methods.<\/li>\n <li>High-dimensional problems.<\/li>\n <li>Stochastic optimization algorithms.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000085_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\">\n <summary>Advanced Modeling<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Continuum mechanics. General conservation laws.<\/li>\n <li>Conservation laws for Newtonian fluids. Main models of fluid dynamics. Non-dimensionalization.<\/li>\n <li>Incompressible ideal flows. Irrotational and potential flows.<\/li>\n <li>Incompressible viscous flows. Navier-Stokes equations.<\/li>\n <li>Turbulent flows. Closure problem and introduction to main turbulence models.<\/li>\n <li>Equations for beams and stationary plates.<\/li>\n <li>Dynamic models. Stability analysis.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000087_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\" class=\"obligatoria\">\n <summary>Advanced Algebra<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Review of rings, modules, localization, and spectrum.<\/li>\n <li>Noetherian rings and modules.<\/li>\n <li>Integral dependence, finite morphisms, the going-up theorem, and Noether’s lemma.<\/li>\n <li>Discrete valuation rings and Dedekind domains. Finiteness theorem. Resolution of singularities.<\/li>\n <li>Completion, Artin-Rees lemma.<\/li>\n <li>Cryptography and computational ideal theory.<\/li>\n <li>Solving multivariate polynomial systems.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000093_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\" class=\"obligatoria\">\n <summary>Advanced Analysis<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Distributions and Sobolev spaces.<\/li>\n <li>Review of the Fourier transform in Euclidean space and in locally compact abelian groups.<\/li>\n <li>Singular integrals and pseudodifferential operators.<\/li>\n <li>Entire functions and the Paley-Wiener theorem. Uncertainty principles.<\/li>\n <li>Hardy spaces and Nevanlinna theory.<\/li>\n <li>Discrete Fourier analysis, sampling theory, wavelets, and applications to image and signal processing.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000092_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\" class=\"obligatoria\">\n <summary>Advanced Differential and Complex Geometry<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Riemannian manifolds, geodesics, Hopf-Rinow theorem, hypersurfaces.<\/li>\n <li>Topology and curvature, constant curvature manifolds, Hadamard and Chern-Gauss-Bonnet theorems.<\/li>\n <li>Conformal structures.<\/li>\n <li>Complex manifolds. Riemann surfaces.<\/li>\n <li>Analytic and meromorphic functions on manifolds, uniformization theorem.<\/li>\n <li>Divisors and the Riemann-Roch theorem.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000094_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Advanced Dynamical Systems<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Review of basic concepts in discrete and continuous dynamical systems.<\/li>\n <li>Invariant sets. Attractors.<\/li>\n <li>Hamiltonian fields and applications.<\/li>\n <li>Bifurcations: codimension. Chaos and chaotic phenomena.<\/li>\n <li>Ergodicity, recurrence, and mixing. Entropy.<\/li>\n <li>Infinite-dimensional dynamics. Semigroups of linear operators. Applications to evolution equations. Infinite-dimensional attractors.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000097_1S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"I\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Advanced Topology<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Homology.<\/li>\n <li>Cohomology.<\/li>\n <li>Universal coefficient theorems, K\u00fcnneth theorem, and duality.<\/li>\n <li>Notable exact sequences: of the closed subspace, Mayer-Vietoris, and Gysin isomorphism.<\/li>\n <li>Explicit calculations, fixed point theorem.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000096_1S_2025-26\">download<\/a><\/p>\n <\/details>\n <\/div>\n\n <div class=\"column\">\n <h2>Second Semester<\/h2>\n\n <details data-semester=\"II\" data-itinerario=\"I\" data-itinerario-highlight=\"aplicada\">\n <summary>Advanced Extension of Statistics and Data Science<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> English<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Specialized data science concepts to generate machine learning models considering probabilistic and statistical models.<\/li>\n <li>Advanced algorithms focused on big data processing, autoencoders, and clustering techniques.<\/li>\n <li>Advanced unsupervised learning algorithms and applications for massive data with many variables or complex dynamics.<\/li>\n <li>Deep learning techniques for predictive, classification, and reconstruction models. Difference between classical and generative models.<\/li>\n <li>Extension and generalization of deep learning models: models based on physical principles.<\/li>\n <li>Reinforcement learning techniques. Industrial and engineering applications.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000091_2S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"II\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Advanced Extension of Algebra, Geometry, and Arithmetic<\/summary>\n <p>\n 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.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Sheaf cohomology and finiteness theorem.<\/li>\n <li>Duality theory, dualizing sheaf, and the Riemann-Roch theorem.<\/li>\n <li>Sheaf of differentials and calculation of the dualizing sheaf.<\/li>\n <li>Projective, injective, and flat modules; torsion and extensions.<\/li>\n <li>Koszul complexes, regular sequences, and Cohen-Macaulay rings.<\/li>\n <li>Regular rings and Serre’s theorem.<\/li>\n <li>Computational methods in noncommutative algebra: factorization and Ore extensions.<\/li>\n <li>Gr\u00f6bner bases in Poincar\u00e9-Birkhoff-Witt rings, quantum groups, differential and difference algebra, characteristic sets.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000100_2S_2025-26\">download<\/a><\/p>\n <\/details>\n\n <details data-semester=\"II\" data-itinerario=\"II\" data-itinerario-highlight=\"fundamental\">\n <summary>Mathematical Physics<\/summary>\n <p>\n This course explores the intersection between theoretical physics and modern mathematical tools, covering field theories, relativity, and quantum mechanics.\n <\/p>\n <p><strong>Language:<\/strong> Spanish<\/p>\n <h4>Specific Contents<\/h4>\n <ul>\n <li>Newtonian gravitation, spacetime, representation of matter, and geometric formulation.<\/li>\n <li>Minkowski spacetime.<\/li>\n <li>Electromagnetism.<\/li>\n <li>General relativity, stress-energy tensor, and Einstein’s equation.<\/li>\n <li>Introduction to quantum mechanics, observables.<\/li>\n <li>Schr\u00f6dinger equation, spin, and the hydrogen atom.<\/li>\n <\/ul>\n <p><strong>Learning guide:<\/strong> <a href=\"https:\/\/www.upm.es\/gauss\/includes_ajax\/guias\/gestion\/gga_gestionar_descarga_pdf.upm?archivo=GA_30AH_303000099_2S_2025-26\">download<\/a><\/p>\n <\/details>\n <\/div>\n <\/div>\n\n <div class=\"info-container\">\n <p><strong>Note:<\/strong><\/p>\n <ul>\n <li>The final course offerings may vary each academic year.<\/li>\n <li>Some subjects are taught in English. For more information click on the corresponding subject above.<\/li>\n <\/ul>\n <p>According to the presented offering, students must complete:<\/p>\n <ol>\n <li>\n Either <strong>18 credits<\/strong> provided by the 3 courses of the <em>Applied Mathematics Block<\/em> in the first semester:\n <ul>\n <li>Ordinary Differential Equations and Applications.<\/li>\n <li>Partial Differential Equations and Their Numerical Approximation.<\/li>\n <li>Advanced Statistics.<\/li>\n <\/ul>\n or <strong>18 credits<\/strong> provided by the 3 courses of the <em>Fundamental Mathematics Block<\/em> in the first semester:\n <ul>\n <li>Advanced Analysis.<\/li>\n <li>Advanced Algebra.<\/li>\n <li>Advanced Differential and Complex Geometry.<\/li>\n <\/ul>\n <\/li>\n <li>Another <strong>24 credits<\/strong> among the remaining courses, including those not selected above, and external academic internships (maximum 6 ECTS).<\/li>\n <li>Finally, <strong>18 credits<\/strong> for the Master’s Thesis (TFM). <a href=\"https:\/\/matematicas.epes.upm.es\/master\/propuestas-de-tfm\/\" target=\"_blank\">Here<\/a> you may consult the topics proposed this year.<\/li>\n <\/ol>\n<ul>\n <li><a href=\"https:\/\/matematicas.epes.upm.es\/master\/wp-content\/uploads\/2025\/09\/Borrador_memoria_Def_y_anexo.pdf\" target=\"_blank\">Here<\/a> you can consult master’s description document (in spanish).<\/li>\n <\/ul>\n <\/div>\n\n <script>\n \/\/ Get references to the buttons\n const btnAplicada2 = document.getElementById(\"btnAplicada\");\n const btnFundamental2 = document.getElementById(\"btnFundamental\");\n\n \/\/ Function to remove highlight from all details\n function removeHighlight2() {\n document.querySelectorAll(\"details\").forEach(detail => {\n detail.classList.remove(\"highlight\");\n });\n }\n\n \/\/ Applied track button\n btnAplicada2.addEventListener(\"click\", () => {\n removeHighlight2();\n document.querySelectorAll('details[data-itinerario=\"aplicada\"]').forEach(detail => {\n detail.classList.add(\"highlight\");\n });\n });\n\n \/\/ Fundamental track button\n btnFundamental2.addEventListener(\"click\", () => {\n removeHighlight2();\n document.querySelectorAll('details[data-itinerario=\"fundamental\"]').forEach(detail => {\n detail.classList.add(\"highlight\");\n });\n });\n <\/script>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Structure and Requirements of the Master’s Program List of Courses (6 ECTS each) for year 25\/26 Applied Mathematics Block Fundamental Mathematics Block 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 … <a href=\"https:\/\/matematicas.epes.upm.es\/master\/en\/structure\/\">Continue reading <span class=\"meta-nav\">→<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-full-width.php","meta":{"footnotes":""},"class_list":["post-113","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages\/113","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/comments?post=113"}],"version-history":[{"count":21,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages\/113\/revisions"}],"predecessor-version":[{"id":586,"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/pages\/113\/revisions\/586"}],"wp:attachment":[{"href":"https:\/\/matematicas.epes.upm.es\/master\/wp-json\/wp\/v2\/media?parent=113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}