“The first principle is that you must not fool yourself and you are the easiest person to fool.” - Richard Feynman
I have met some incredibly inspiring professors over the years, and I try to live up to their standards (or at least try to).
Courses taught
Mathematics (2024)
Content: Single variable calculus (real functions, limits, derivatives, integrals). Introduction to ordinary differential equations. Introduction to numerical analysis.
Mathematics II (2022-2024)
Content: Multivariable functions. Partial derivatives and gradients. Multiple integration, polar/cylindrical/spherical coordinate systems. Vector fields, divergence and curl. Line and surface integrals. Theorems of Green, Stokes and Gauss. Introduction to ordinary differential equations.
Calculus (2022-2024)
Content: Basic notions of Complex numbers. Real functions. Limits and derivatives. Indefinite integrals. Definite integrals and the Fundamental Theorem of Calculus. Successions and series, convergence tests. Power series and the Taylor expansion.
Discrete Mathematics and Algebra (2022-2023)
Content (Discrete Mathematics): Diophantine equations. Modular arithmetic. Combinatorics. Graph theory.
Content (Algebra): Systems of equations and matrices. Vector spaces. Linear maps. Diagonalization.
Mathematical Methods III (2018-2019)
Content: Introduction to the Complex numbers. \(\mathbb{C}\)-valued functions and the Cauchy-Riemann equations. Complex integration and the Residue Theorem. Analytic continuation, the Gamma and Riemann zeta functions.
Random materials I have created over the years
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String Theory, extension: An introduction to the AdS/CFT correspondence (UvA, 2018-2019)
After passing the String Theory, extension course, I discussed with the two lecturers the possibility of creating some lecture notes for said course, which could be used in future years as the official course bibliography. They thought it was a great idea and we worked on in for some time, with this being the final product.
They cover the basics of the AdS/CFT correspondence (the first, yet only realization of the concept of Holography), establishing the necessary ingredients on the AdS side (Anti-de Sitter geometry and symmetries, Fefferman-Graham coordinate expasion, ...) and those on the CFT side (Supersymmetry, \(\mathcal{N}=4\) Super Yang-Mills, ...). The final chapters are devoted to some specific applications such as black hole entropy calculations and renormalization group flows.
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Cosmology lecture notes - in Spanish (UCM, 2017)
I wrote these notes during the Cosmology course from the last year of the Physics bachelor. They contain everything (including some suggested problems) covered in the course, namely the derivation of the Friedman-Lemaître-Robertson-Walker metric from the Cosmological Principle, cosmological kinematics and dynamics, and the problems with the current (\(\Lambda\)CDM) Cosmological Model. Standard Differential Geometry knowledge is assumed from the start. It includes proposed exercises, some of which are solved at the end.
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Collection of Quantum Field Theory solved exercises - in Spanish (UCM, 2016-2017)
The best way to learn something is by carrying out a project where you need it. Following that principle, I decided to type in \(\LaTeX\) all the exercises we were supposed to hand in during the Quantum Field Theory course (and some extra ones, although I did not include their solution there). The exercises are, mostly, hands on calculations which the professor skipped in the lectures, meant for us to practice some common techniques and mathematical trickery used in this subject.