Mathematics and Engineering Applications

Interactive visualisations, presentation series, and explorations in pure mathematics, cryptography, and coding theory.

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Cryptography

Interactive slide decks covering modern cryptographic systems end-to-end — from the underlying mathematics through NIST standards to SystemVerilog RTL and Python implementations.

Open the Cryptography Series

#	Topic	Slides	Status
01	Finite Field Arithmetic for Cryptography	16	✅ Complete
02	Elliptic Curve Cryptography (ECC)	38	✅ Complete
03	AES — Design & Implementation	28	✅ Complete
04	Hash Functions & MACs	29	✅ Complete
05	Public Key Cryptography (RSA, DH)	28	✅ Complete
06	Digital Signatures (ECDSA, EdDSA, Schnorr)	25	✅ Complete
07	Post-Quantum Cryptography	21	✅ Complete
08	Fully Homomorphic Encryption (FHE)	39	✅ Complete
09	Side-Channel Attacks & Countermeasures	22	✅ Complete
10	Crypto Hardware Accelerator Design	38	✅ Complete
11	Zero-Knowledge Proofs	28	✅ Complete
12	Secure Multi-Party Computation	27	✅ Complete
13	TLS 1.3 Handshake	25	✅ Complete

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Coding Theory

Interactive slide decks on error detection and error correction — from parity bits and Hamming codes through Reed-Solomon, turbo codes, LDPC, and polar codes.

Open the Coding Theory Series

#	Topic	Slides	Status
01	Foundations of Coding Theory	~25	✅ Complete
02	Parity Checks & Simple Codes	~22	✅ Complete
03	Hamming Codes	~25	✅ Complete
04	CRC & Error Detection	~22	✅ Complete
05	Linear Block Codes	~24	✅ Complete
06	Convolutional Codes & Viterbi Decoding	~25	✅ Complete
07	BCH Codes	~22	✅ Complete
08	Reed-Solomon Codes	~28	✅ Complete
09	Turbo Codes & Iterative Decoding	~24	✅ Complete
10	LDPC Codes	~24	✅ Complete
11	Fountain & Rateless Codes	~20	✅ Complete
12	Polar Codes	~22	✅ Complete
13	Coding in Practice — Real-World Systems	~26	✅ Complete

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Quantitative Finance (a companion to Paul Wilmott's Paul Wilmott Introduces Quantitative Finance)

A fourteen-deck interactive companion to the John Wiley & Sons textbook by Paul Wilmott (2nd edition, 2007) — the standard derivative-pricing reference written from the trader's point of view. Decks 01–05 introduce the products, markets and the random walk; decks 06–10 develop the Black–Scholes world and its exotic extensions; decks 11–12 cover fixed income and interest-rate models; decks 13–14 close with risk management and the numerical methods that price what closed forms cannot. Every deck has at least one in-browser interactive widget.

Open the Quantitative Finance Series

#	Topic	Coverage	Status
01	Products and Markets	Equities, dividends, splits, commodities, currencies, indices, time value of money, fixed-income basics, forwards and futures, no-arbitrage — interactive forward-pricing calculator	✅ Complete
02	Derivatives	Calls, puts, payoff diagrams, put–call parity, binaries, bull/bear spreads, straddles, strangles, butterflies, condors, calendars — interactive option-strategy composer	✅ Complete
03	The Binomial Model	One-step and multi-step trees, risk-neutral probability, delta hedging, the continuous-time limit — interactive binomial-tree pricer	✅ Complete
04	The Random Behavior of Assets	Returns, timescales, drift & volatility, the lognormal random walk, the Wiener process — interactive GBM path simulator	✅ Complete
05	Elementary Stochastic Calculus	Markov & martingale properties, quadratic variation, Brownian motion, stochastic integration, SDEs, Itô's lemma — interactive Brownian-motion + quadratic-variation visualiser	✅ Complete
06	The Black–Scholes Model	The special portfolio, elimination of risk by delta hedging, no arbitrage, the BS PDE, assumptions, boundary & final conditions, PDE solution methods — interactive option-value surface viewer	✅ Complete
07	The Greeks and Hedging	Formulæ for calls, puts, binaries; Δ, Γ, Θ, Vega, ρ; implied volatility; the classification of hedging types — interactive Greeks explorer	✅ Complete
08	Overview of Volatility Modeling	Actual, historical, implied & forward volatility; GARCH; range-based estimators; skews and smiles; deterministic, stochastic and uncertain vol — interactive implied-vol solver and smile/skew explorer	✅ Complete
09	Exotic and Path-dependent Options	The option taxonomy: time and path dependence, dimensionality, order, embedded decisions. Barriers, Asians, lookbacks, compounds, Parisians — interactive barrier-option Monte Carlo	✅ Complete
10	Multi-asset Options	Multidimensional lognormal walks, the correlation matrix, Cholesky, exchange options (Margrabe), basket options, correlation vs cointegration — interactive correlated-GBM scatter and basket-payoff explorer	✅ Complete
11	Fixed-income Products, Yield and Swaps	Zero-coupon and coupon bonds, money-market account, FRAs, repos, STRIPS, day-count conventions, YTM, duration, convexity, bootstrapping forward rates, vanilla IRS — interactive bond-yield / duration calculator	✅ Complete
12	Interest Rate Models	Stochastic short rates (Vasicek, CIR, Ho–Lee, Hull–White), the bond-pricing equation, the market price of risk, yield-curve fitting; the HJM forward-rate framework and the BGM LIBOR market model — interactive short-rate path simulator	✅ Complete
13	Portfolio Management, VaR & Risk	The Kelly criterion, diversification, Modern Portfolio Theory & the efficient frontier, CAPM, VaR for assets and derivatives, credit risk & the Merton model, copulas, CrashMetrics, the great derivatives disasters — interactive efficient-frontier + VaR explorer	✅ Complete
14	Numerical Methods	Finite differences, Monte Carlo, numerical integration; the explicit FD scheme for BS, Monte Carlo for European and American options (Longstaff–Schwartz), low-discrepancy sequences — interactive Monte Carlo + finite-difference pricer	✅ Complete

Book is by Paul Wilmott, John Wiley & Sons (2nd edition, 2007), ISBN 978-0-470-31958-1. The companion is independent and unofficial — all credit for the underlying exposition belongs to the author.

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Mathematics for Machine Learning (a companion to Deisenroth, Faisal & Ong's Mathematics for Machine Learning)

A twelve-deck interactive companion to the Cambridge University Press textbook by Marc Peter Deisenroth, A. Aldo Faisal & Cheng Soon Ong (2020) — the standard "honest mathematics, written for ML readers" text. Decks 01–07 develop the foundations (linear algebra, geometry, matrix decompositions, calculus, probability, optimisation); decks 08–12 use them to derive the four canonical ML problems (regression, PCA, GMM, SVM). Every deck has at least one in-browser interactive widget.

Open the Mathematics for Machine Learning Series

#	Topic	Coverage	Status
01	Introduction and Motivation	The four pillars, the four foundations, and an interactive dependency graph of the book	✅ Complete
02	Linear Algebra	Systems, matrices, Gaussian elimination (interactive), vector spaces, basis & rank, linear maps, affine	✅ Complete
03	Analytic Geometry	Norms, inner products, angles, orthonormal bases, orthogonal projection (interactive), rotations	✅ Complete
04	Matrix Decompositions	Determinant, trace, eigendecomposition, spectral theorem, Cholesky, SVD, Eckart–Young, interactive SVD compressor	✅ Complete
05	Vector Calculus	Gradient, Jacobian, matrix-derivative identities, chain rule, backprop, autodiff, Hessian, gradient-descent visualiser	✅ Complete
06	Probability and Distributions	Sum/product/Bayes, expectation & variance, multivariate Gaussian (conditioning), conjugacy, exponential family — with interactive Bayes' theorem and bivariate Gaussian	✅ Complete
07	Continuous Optimisation	Gradient descent + momentum, Lagrange multipliers (interactive), KKT, convexity, LP/QP, the Lagrange dual	✅ Complete
08	When Models Meet Data	ERM, bias-variance, regularisation as a prior, cross-validation, MLE vs MAP, directed graphical models — interactive polynomial-fitting demo	✅ Complete
09	Linear Regression	Least squares as projection, ridge = MAP, Bayesian linear regression, posterior predictive, feature maps — interactive Bayesian regression playground	✅ Complete
10	Dimensionality Reduction with PCA	Two equivalent objectives, eigendecomposition of covariance, PCA via SVD, high-D dual trick, probabilistic PCA — interactive 2D PCA	✅ Complete
11	Density Estimation with GMMs	Mixture density, latent variables, EM as a variational lower bound, responsibilities, M-step formulae — interactive 2D EM animator	✅ Complete
12	Classification with SVMs	Max margin, soft margin / hinge, Lagrange dual, support vectors, kernel trick (linear / poly / RBF) — interactive 2D kernel SVM	✅ Complete

Book is freely available as a PDF at mml-book.github.io. The companion is independent and unofficial — all credit for the underlying exposition belongs to Deisenroth, Faisal & Ong.

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Mathematical Basic Training for Science (a companion to R. Shankar's Basic Training in Mathematics)

A self-contained curriculum in the core mathematical methods used across the physical sciences — from single-variable calculus through complex analysis, vector calculus, linear algebra and differential equations. Every topic is an interactive single-page app in the same Plotly.js + dark-theme style as the rest of this collection.

#	Topic	Coverage	Status
01	Differential Calculus of One Variable	Limits, derivative as a limit, rules of differentiation, L'Hôpital's rule, extrema & concavity, differentials & linearisation	✅ Complete
02	Integral Calculus	Riemann sums, fundamental theorem, substitution, parts, partial fractions, improper integrals, Gaussian & Gamma	✅ Complete
03	Calculus of Many Variables	Scalar fields, partial derivatives, gradient, directional derivatives, Lagrange multipliers, Jacobians, solid angle	✅ Complete
04	Infinite Series	Sequences, partial sums, convergence tests, power series & radius of convergence, asymptotic series	✅ Complete
05	Complex Numbers & Functions of a Complex Variable	Argand arithmetic, De Moivre, Euler, Cauchy–Riemann, singularities, residue theorem, Riemann sphere	✅ Complete
06	Vector Calculus	Vector fields, gradient, divergence, curl, line & surface integrals, Green's/Stokes'/divergence theorems, electromagnetism	✅ Complete
07	Matrices & Determinants	Matrix operations, determinants, inverses, linear transformations, eigenvalues & eigenvectors — paired with Eigenvalue_Explorer	✅ Complete
08	Linear Vector Spaces	Axioms, span & independence, inner products, Gram-Schmidt, operators, dual space, tensor products, eigenvalue problem, function spaces & Fourier integrals, Hilbert & Banach spaces — six-repo series: History, Basics, Advanced, DSP & Audio, Quantum, ML	✅ Complete
09	Differential Equations	Direction fields, damped oscillator, linear systems, Frobenius/Bessel/Legendre, wave & heat equations, Green's functions	✅ Complete

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Mathematics

Interactive single-page visualisations built with Plotly.js and vanilla HTML/CSS/JS — no build step, no dependencies to install.

Analysis & Calculus

Project	Description
Taylor Series Visualisation	Interactive Plotly.js visualisations of Taylor polynomial approximations for 20 functions — convergence regions, adjustable order, dark theme
Complex Taylor Series Visualisation	Domain colouring and 3D magnitude surfaces for Taylor series of 12 complex functions — split-view comparison, presets, convergence boundaries
Fourier Series Visualisation	Build periodic waveforms term by term — square, sawtooth, triangle and more with Gibbs phenomenon, frequency spectra, and adjustable harmonics
Curvature Visualiser	Osculating circles, curvature plots, and evolutes for parametric curves — ellipses, spirals, cardioids, Lissajous, and more
Vector Calculus Primer	Intuition-first interactive primer (10 tabs): vector fields, gradient, divergence, curl, line integrals, fundamental theorems, vector algebra, del identities, electromagnetism applications, and 3D vector fields
Differential Calculus Explorer	Limits with ε-δ strip, secant→tangent convergence, differentiation rules, L'Hôpital, critical/inflection detection, tangent linearisation
Integral Calculus Explorer	Riemann sums (left/right/midpoint/trapezoidal/Simpson), FTC dual-plot, substitution, parts (LIATE), partial fractions, improper integrals, Gamma & Beta
Multivariable Calculus Explorer	3D surfaces, partial slices, gradient quiver on contours, directional derivatives, Lagrange multipliers, Jacobians, solid angle on the sphere
Infinite Series Explorer	Sequences with ε bands, partial sums, convergence tests (ratio/root/integral/comparison/alternating), radius of convergence, Stirling asymptotics

Complex Analysis

Project	Description
Conformal Mapping Gallery	Interactive gallery of complex mappings — z², eᶻ, 1/z, Möbius, Joukowski airfoil — with grid warping, domain colouring, and animation
Riemann Zeta Visualisation	Domain colouring of ζ(s) on the critical strip — explore non-trivial zeros, the critical line, and the functional equation
Complex Analysis Explorer	Argand arithmetic, roots of unity, Euler's formula, Cauchy–Riemann surfaces, singularity classification, residue theorem with contour, stereographic Riemann sphere

Linear Algebra

Project	Description
Linear Transformation Visualiser	See how 2×2 matrices warp grids, circles, and vectors — eigenvalue/eigenvector overlays, SVD decomposition, animated transitions
Eigenvalue Explorer	Drag matrix entries and watch eigenvalues move in the complex plane — supports 2×2 to 4×4, trails, stability colouring, and characteristic polynomials

Matrix Methods in Engineering

A three-deck series on the matrix structures that engineers actually test for — symmetry, definiteness, unitarity — what each one guarantees about a real system, and what happens when you take the whole apparatus to a real design problem. Built in the same interactive slide-deck style as the companion series above, with live canvas widgets and a long-form PDF alongside each deck.

Project	Description
Network Parameters — S, Z and Y	One linear operator, three coordinate systems: reciprocity as transpose symmetry, passivity as positive semidefiniteness, losslessness as unitarity. Poles and zeros as eigenvalues of the state matrix, causality and Kramers–Kronig, Hamiltonian passivity enforcement, the Smith chart as a Möbius map. Interactive two-port scattering explorer, pole–zero explorer, and bilinear map
Digital Filters — State, Structure and Stability	The same structures in discrete time: eigenvalues of the state matrix and the unit circle, Wiener–Hopf with Toeplitz Hermitian PSD autocorrelation, the eigenfilter as a Rayleigh quotient, paraunitary filter banks and perfect reconstruction. Interactive pole explorer, eigenfilter bowl, and a paraunitary rotation you can break
Equalisation in High-Speed Serial Links	The two above, applied end to end: one 28 GBd backplane channel taken from S-parameters through the pulse response and every equaliser block to a noise budget that lands 1.36 dB short — then seven remedies priced in dB, showing the board is worth five times what more equalisation is worth. Interactive via-stub explorer and a full equaliser chain with live budget, Q and BER

Vector Spaces

A six-repo series on vector spaces — from their 19th-century origins through the abstract axioms, advanced functional analysis, and the decompositions that drive modern audio, quantum mechanics, and machine learning. All built in the same DM Sans + JetBrains Mono + Playfair Display dark style, with canvas-based interactive visualisations.

Project	Description
Vector Spaces — A History	Interactive timeline 1799–1932, portrait cards, force-directed concept map, idea threads, and period quotations — from Hamilton's quaternions and Grassmann's Ausdehnungslehre through Peano's axioms to Hilbert, Banach, and von Neumann
Vector Spaces — The Basics	Ten tabs: axioms with live check, span & linear combinations, independence & rank, basis building, subspaces with Grassmann's formula, inner products, animated Gram-Schmidt, p-norm unit balls, linear maps, and change of basis
Vector Spaces — Advanced Topics	Dual space (covectors as level sets), quotient cosets, direct sum decomposition, tensor products, ℓ² and Hilbert space with Parseval, Banach Lᵖ, operator norm & condition number, spectral theorem, Riesz representation
Vector Spaces — DSP for Sound, Audio & Music	Twelve tabs covering signal-as-vector (with audio playback), DFT, DCT/MDCT, STFT, wavelets, mel/MFCC, SVD denoising, PCA/KL, NMF, ICA, matching pursuit, CQT — every audio transform viewed as a change of basis
Vector Spaces in Quantum Mechanics	State as unit vector, Bloch sphere with presets, observables & eigenprojectors, measurement sampling, unitary time evolution (animated), Schmidt rank & entanglement, density matrices inside the Bloch ball
Vector Spaces in Machine Learning	Features as vectors with live linear classifier, word-embedding analogy parallelograms, kernel trick with RBF decision surface, PCA projections, attention as soft retrieval, contrastive geometry on the hypersphere, latent interpolation (lerp vs slerp)

Dynamical Systems & Control

Project	Description
ODE Phase Portraits	Vector fields and trajectories for 2D dynamical systems — Van der Pol, Lotka-Volterra, Duffing, with nullclines and fixed-point classification
Laplace Pole-Zero Explorer	Interactive s-plane editor — drag poles and zeros, watch impulse response and Bode plots update live with stability classification
Differential Equations Explorer	Direction fields with RK4 trajectories, damped driven oscillator, 2×2 phase portraits, Bessel/Legendre/Airy, wave & heat equations, Green's function convolution

Abstract Algebra

Project	Description
Group Theory Visualiser	Cayley graphs, multiplication tables, subgroup lattices, and symmetry playground for cyclic, dihedral, symmetric, and quaternion groups
Galois Fields in Hardware	Interactive Reveal.js presentation on Galois Field (GF(2^n)) theory for hardware engineers — finite field arithmetic, irreducible polynomials, and applications in CRC, LFSR, AES, and error-correcting codes

Topology & Geometry

Project	Description
Topology & Knot Theory Visualiser	3D surfaces (Möbius strip, Klein bottle, torus), Euler characteristic calculator, knot explorer, and fundamental polygon identification
Differential Geometry	Gaussian curvature on surfaces, geodesic tracer, parallel transport with holonomy, and interactive Gauss-Bonnet theorem
Hyperbolic Geometry	Poincaré disk and upper half-plane models, hyperbolic tilings {p,q}, isometry explorer, and hyperbolic triangle angle-sum

Number Theory

Project	Description
Prime Spirals	Ulam, Sacks, and Archimedes spirals with primes highlighted — configurable colouring by factorisation, twin primes, and prime gaps

Probability & Statistics

Project	Description
Central Limit Theorem	Sample from any distribution and watch the sum converge to a Gaussian — animated histogram, Q-Q plot, and statistics panel
Random Walk Visualisation	1D and 2D random walks, Brownian motion, and Lévy flights — diffusion envelopes, displacement statistics, and heat equation connection
Markov Chain Visualisation	State diagram editor, animated simulation, stationary distribution convergence, state classification, and matrix analysis

Information Theory

Project	Description
Information Theory	Shannon entropy, mutual information Venn diagrams, channel capacity (BSC, BEC), Huffman coding, and KL divergence

Numerical Analysis

Project	Description
Numerical Methods	Root finding (bisection, Newton, secant), numerical integration, ODE solvers (Euler, RK4), and polynomial interpolation — all animated step-by-step

Control Theory

Project	Description
Control Theory	PID tuning, root locus, Bode plots, Nyquist diagrams, and state-space analysis with real-time simulation

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Historical Mathematicians

88 interactive presentations (17 slides each, all ✅ complete) on the lives, discoveries, and influence of history's greatest mathematicians — covering Bell's Men of Mathematics, Struik's A Concise History, and Stillwell's Mathematics and Its History.

Open the Historical Mathematicians Series

#	Mathematician	#	Mathematician	#	Mathematician
01	Pythagoras	31	Leibniz	61	Kummer
02	Zeno of Elea	32	De Moivre	62	Sylvester
03	Eudoxus	33	Taylor	63	Stokes
04	Euclid	34	Stirling	64	Cayley
05	Archimedes	35	Maclaurin	65	Weierstrass
06	Apollonius	36	The Bernoullis	66	Boole
07	Diophantus	37	Euler	67	Hermite
08	Brahmagupta	38	d'Alembert	68	Kronecker
09	Al-Khwarizmi	39	Bayes	69	Venn
10	Omar Khayyám	40	Lambert	70	Riemann
11	Alhazen	41	Lagrange	71	Dedekind
12	Buridan	42	Laplace	72	Lie
13	Oresme	43	Legendre	73	Klein
14	Fibonacci	44	Monge	74	Heaviside
15	Pacioli	45	Fourier	75	Kovalevskaya
16	Del Ferro	46	Germain	76	Cantor
17	Tartaglia	47	Poncelet	77	Poincaré
18	Cardano	48	Gauss	78	Hilbert
19	Ferrari	49	Poisson	79	Lebesgue
20	Bombelli	50	Green	80	Ramanujan
21	Viète	51	Cauchy	81	Noether
22	Stevin	52	Lobachevsky	82	Von Neumann
23	Napier	53	Bolyai	83	Gödel
24	Harriot	54	Abel	84	Turing
25	Descartes	55	Dirichlet	85	Erdős
26	Fermat	56	Jacobi	86	Grothendieck
27	Pascal	57	De Morgan	87	Mandelbrot
28	Wallis	58	Hamilton	88	Conway
29	Huygens	59	Galois
30	Newton	60	Liouville

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Historical Physicists

37 interactive presentations (17 slides each, all ✅ complete) on the lives, experiments, and theories of history's greatest physicists — from Galileo's telescopic revolution through Penrose's singularity theorems, covering the scientific revolution, classical electromagnetism, signal processing pioneers, the quantum revolution, and modern cosmology.

Open the Historical Physicists Series

#	Physicist	#	Physicist	#	Physicist
01	Galileo	13	Lodge	25	Heisenberg
02	Kepler	14	Hertz	26	Pauli
03	Huygens	15	Tesla	27	Dirac
04	Newton	16	J.J. Thomson	28	Oppenheimer
05	Ampère	17	Curie	29	Fermi
06	Faraday	18	Planck	30	Bardeen
07	Kelvin	19	Rutherford	31	Feynman
08	Tait	20	Einstein	32	Gell-Mann
09	Maxwell	21	Bohr	33	Bell
10	Boltzmann	22	Bromwich	34	Penrose
11	Gibbs	23	Born	35	Higgs
12	FitzGerald	24	Schrödinger	36	Deutsch
				37	Hawking

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License

Educational use. Code examples provided as-is. Standards references are to publicly available NIST, IETF, and ISO documents.