My Notes

These are my notes for some of my courses at the University of Edinburgh & the University of Oxford.

Clicking on the course name takes you to the GitHub repository folder on which all the notes for the corresponding course can be found.

Clicking on the arrow (▸) next to the course name will provide links to access each of the individual chapters for the course.

University of Edinburgh

Year 2

Fundamentals of Pure Mathematics 👥

Introduction to Algorithms and Data Structures 🌳

• Asymptotics, Sorting, Core Data Structures and Graphs
• Dynamic Programming, Parsing and NP-Completeness

Year 3

Honours Algebra 💍

• Chapter 1: Fields and Vector Spaces
• Chapter 2: The "Morphisms" and Representing Matrices
• Chapter 3: Abstract Linear Mappings and Change of Basis Matrices
• Chapter 4: Rings, Ideals and Subrings
• Chapter 5: Factor Rings, The First Isomorphism Theorem and Modules
• Chapter 6: Determinants and Multilinear Forms
• Chapter 7: Eigenvalues, Triangularisation and Diagonalisation
• Chapter 8: Inner Product Spaces and Orthogonal Projections
• Chapter 9: Adjoint Endomorphisms and The Spectral Theorem
• Chapter 10: The Jordan Normal Form

Honours Analysis ♾️

• Chapter 1: Real Numbers and Sequences
• Chapter 2: Bolzano-Weierstrass Theorem, Cauchy Sequences and Series
• Chapter 3: Continuity and Uniform Convergence of Sequences
• Chapter 4: Uniform Convergence of Series
• Chapter 5: Power Series
• Chapter 6: Lebesgue Integrable Functions
• Chapter 7: Lebesgue Integrability of Series and the Riemann Integral
• Chapter 8: Lebesgue Integrability Over Intervals and the Fundamental Theorem of Calculus
• Chapter 9: Fatoux Lemma and the Dominated Convergence Theorem
• Chapter 10: The Space of L^2 Functions
• Chapter 11: Fourier Series

Honours Differential Equations 🌀

• Chapter 1: Solving Linear Systems
• Chapter 2: Fundamental Matrices
• Chapter 3: Solving Nonhomogeneous Systems
• Chapter 4: Phase Portraits and System Stability
• Chapter 5: Lyapunov Functions
• Chapter 6: Introduction to Fourier Series
• Chapter 7: Solving PDEs
• Chapter 8: Introduction to Sturm Liouville Theory
• Chapter 9: The Laplace Transform (Incomplete)

Foundations of Natural Language Processing 💬

• Chapter 1: Intro to NLP, Ambiguity and Working with Corpora
• Chapter 2: N-Gram Models and Smoothing Techniques
• Chapter 3: Further Smoothing, Noisy Channel Model and Naive Bayes for Text Classification
• Chapter 4: Logistic Regression for Text Classification, Morphological Parsing and POS Tagging
• Chapter 5: Syntactic Parsing, CKY and PCFGs
• Chapter 6: Evaluating Parsing, Improving Vanilla PCFGs, Dependency Parsing and Semantics from Syntax
• Chapter 7: Semantic Role Labelling, Word Sense Disambiguation and Distributional Semantics
• Chapter 8: Word2Vec, ANNs for NLP and Discourse Coherence

Introductory Applied Machine Learning 🖥️

• Chapter 1: Basic Math and Naive Bayes
• Chapter 2: Decision Trees
• Chapter 3: Linear and Logistic Regression
• Chapter 4: Optimisation, Regularisation and SVMs
• Chapter 5: K Nearest Neighbours
• Chapter 6: Gaussian Mixture Models and K Means
• Chapter 7: PCA and Hierarchical Clustering
• Chapter 8: Introduction to Artificial Neural Networks

Year 4

Galois Theory 📐

• Chapter 1: Introduction to Conjugacy
• Chapter 2: Group Actions, Rings and Fields
• Chapter 3: Polynomials
• Chapter 4: Field Extensions and Homomorphisms Over Fields
• Chapter 5: Degree of Extensions, Finite Extensions and Ruler + Compass Constructions
• Chapter 6: Splitting Fields and the Galois Group
• Chapter 7: Normal and Separable Extensions
• Chapter 8: The Fundamental Theorem of Galois Theory
• Chapter 9: Solvability by Radicals
• Chapter 10: Galois Correspondence for Finite Fields
• Galois Theory Cheatsheet
• Original Notes by Tom Leinster

Group Theory 🌗

• Chapter 1: Introduction to Groups
• Chapter 2: The Isomorphism Theorems
• Chapter 3: The Free Group
• Chapter 4: Group Actions and the Sylow Theorems
• Chapter 5: The Fundamental Theorem of Finite Abelian Groups
• Chapter 6: The Fundamental Theorem of Finitely Generated Abelian Groups
• Chapter 7: The Alternating Group
• Chapter 8: Composition Series and the Jordan-Hölder Theorem
• Chapter 9: Solvable Groups and the Derived Series

Introduction to Partial Differential Equations 🌊

• Chapter 1: Partial Differential Equations and the Transport Equation
• Chapter 2: The Heat Equation: Solution via Fourier Series
• Chapter 3: The Fundamental Solution to the Heat Equation and the Weak Maximum Principle
• Chapter 4: Harmonic Functions and The Fundamental Solution to the Poisson Equation
• Chapter 5: Green's Function, Harnack's Inequality and Liouville's Theorem
• Chapter 6: The Wave Equation (D'Alembert's and Kirchhoff's Formulae)

Variational Calculus 🪐

• Chapter 0: Prequel: Several Variable Calculus and Analysis
• Chapter 1: Geodesics: Extremals of the Arclength Functional
• Chapter 2: The Euler-Lagrange Equations
• Chapter 3: Newton's Equation and Galilean Gravity
• Chapter 4: Conservative Forces and Hamilton's Principle of Least Action
• Chapter 5: Diffeomorphisms and Noether's Theorem (Simple Version)
• Chapter 6: The General Noether's Theorem and the Hamiltonian Formalism
• Chapter 7: Canonical Transformations and Integrable Systems
• Chapter 8: The Isoperimetric Problem: Lagrange Multipliers with Functional Constraints
• Chapter 9: Holonomic Constraints: Lagrange Multipliers with Function Constraints
• Chapter 10: The Euler-Lagrange Equations in Higher Dimensions

Machine Learning and Pattern Recognition 🤖

• Chapter 1: Linear Regression and Regularisation
• Chapter 2: Evaluating Models and Multivariate Gaussians
• Chapter 3: Classification: Linear Regression, Bayesian Models and Logistic Regression
• Chapter 4: Bayesian Linear Regression
• Chapter 5: Gaussian Processes
• Chapter 6: Kernels for Gaussian Processes and Softmax Classification
• Chapter 7: Neural Networks
• Chapter 8: Autoencoders, PCA and SVD
• Chapter 9: Bayesian Logistic Regression and the Laplace Approximation
• Chapter 10: Importance Sampling and Variational Inference with KL-Divergence

Natural Language Understanding, Generation and Machine Translation 🗯️

• Chapter 1: N-Grams for Language Modelling and Translation Linear
• Chapter 2: Neural Language Models
• Chapter 3: Encoder-Decoder Models: RNNs and Transformers
• Chapter 4: Word Embeddings and BERT
• Chapter 5: Prompting and Evaluating Machine Translation
• Chapter 6: Open Vocabulary Models and Low-Resource Machine Translation
• Chapter 7: Summarisation with Neural Models
• Chapter 8: Supervised and Unsupervised Neural Parsing
• Chapter 9: Summarising Movies

University of Oxford

Introduction to Representation Theory 🧩

• Chapter 1: Representations of Finite Groups
• Chapter 2: Representations and Modules
• Chapter 3: Decomposing Semisimple Rings
• Chapter 4: Generating New Representations
• Chapter 5: Character Theory
• Chapter 6: Induced Representations
• Chapter 7: Algebraic Integers and Burnside’s Theorem

Networks 🕸️

• Chapter 1: Randomness and Matrices
• Chapter 2: Exploring Properties of Networks
• Chapter 3: Community Detection and Katz Centrality for Dynamic Networks
• Chapter 4: Non-Linear Evolving Network Models
• Chapter 5: Centrality in Continuous Time & Dynamics on Networks
• Chapter 6: Discrete Random Walks on Networks
• Chapter 7: Scaling Properties of Networks & Decomposing Networks