Sequences, series, and convergence: tests and formal proofs · Multivariable calculus: partial derivatives, gradient, and Hessian matrix · Multiple integrals: double, triple, and change of variables (Jacobian) · Vector calculus: divergence, curl, Green’s, Stokes’, and Gauss’ theorems · Fourier series and Laplace transforms — introduction and applications

Vector spaces, subspaces, span, bases, and dimension · Linear transformations, matrices, rank, and nullity theorem · Eigenvalues, eigenvectors, diagonalisation, and Jordan normal form · Inner product spaces, orthogonality, and Gram-Schmidt process · Applications: Google PageRank, PCA, computer graphics, and quantum computing

Probability spaces, random variables, and distribution functions · Expectation, variance, moment generating functions, and characteristic functions · Central Limit Theorem and Law of Large Numbers — proofs and applications · Bayesian inference: prior, posterior, credible intervals, and MCMC introduction · Regression analysis, ANOVA, and non-parametric statistical tests

Mathematical logic: propositions, quantifiers, and proof strategies · Set theory: operations, power sets, and Cartesian products · Graph theory: paths, trees, planarity, colouring, and network flow · Combinatorics: permutations, combinations, pigeonhole, inclusion-exclusion · Number theory: modular arithmetic and RSA cryptography basics

Root-finding: bisection, Newton-Raphson, and secant methods · Numerical integration: trapezoidal, Simpson’s, and Gaussian quadrature · ODE solvers: Euler, Runge-Kutta methods, and stability analysis · Mathematical modelling: SIR epidemiological models and optimisation · Error analysis, convergence rates, and computational complexity

Linear algebra for ML: SVD, PCA, and matrix factorisation · Calculus for optimisation: gradient descent and backpropagation mathematics · Probability for Bayesian networks and probabilistic graphical models · Statistics for experimental design, A/B testing, and causal inference · Mathematical foundations of neural networks and deep learning

Apply advanced calculus, linear algebra, and real analysis to real-world problems · Implement numerical methods and mathematical models computationally · Use probability and statistics at a professional research level · Understand the mathematical foundations of AI and data science · Achieve a credential equivalent to first-year university mathematics distinction

Complete all 6 modules, pass a university-level proctored examination (minimum 75%), and submit an original mathematical modelling or research project with a written report.

Data Scientist, Machine Learning Engineer, Quantitative Analyst, Research Mathematician, Actuary, Financial Modeller, Aerospace Engineer, Biostatistician, and academic research across all STEM disciplines.