Mathematics Hub

Mathematics provides the formal language and foundational structures for science, engineering, and computing. This hub organizes Wikantik's mathematical content as a course sequence — the order an undergraduate curriculum actually follows — from foundations and proof-writing through the calculus and algebra sequences to analysis, probability, and the applied toolkit.

Foundations, Logic, and Proof

The bedrock of mathematical reasoning and the formal systems that underpin computation.

• Proof Techniques and Mathematical Reasoning — The transition course: direct, contrapositive, contradiction, induction, and epsilon-delta
• Set Theory and Logic — ZFC axioms, cardinality, and the standard foundation of modern math
• Propositional Logic — The logic of "and", "or", and "not"; the basis of digital circuits
• Predicate Logic — First-order logic, quantifiers, and the foundation of relational databases
• Infinity Mathematics — Comparing different sizes of infinity and the continuum hypothesis
• Symbolic Logic — Formal manipulation of symbols and its role in automated reasoning
• Temporal Logic — Reasoning about propositions qualified in terms of time; critical for system verification
• Modal Logic — The logic of necessity and possibility

The Calculus Sequence

Change, accumulation, and infinite processes — the standard three-semester arc plus differential equations.

• Differential Calculus — Derivatives, Taylor approximation, Jacobians and Hessians
• Integral Calculus — The Riemann integral, the Fundamental Theorem, techniques, and improper integrals
• Sequences and Series — Convergence tests, power series, and Taylor expansions with error bounds
• Multivariable and Vector Calculus — Multiple integrals, gradients, and Green's, Stokes', and the Divergence theorems
• Ordinary Differential Equations — First- and second-order equations, systems via eigenvalues, Laplace transforms
• Partial Differential Equations — Heat, wave, and Laplace equations; separation of variables and characteristics
• Fourier Analysis — Fourier series and transforms, convolution, and the FFT
• Calculus Refresh for CS — Targeted calculus for software engineers and ML practitioners

The Algebra Sequence

Mathematical structures and the rules for manipulating symbols within them.

• Linear Algebra — Vectors, matrices, and linear transformations; the computational engine of ML and graphics
• Abstract Algebra — Groups, rings, and fields: the math behind cryptography and error correction
• Group Theory and Symmetry — Formalizing symmetry and its applications in physics and chemistry
• Galois Theory and Field Extensions — The Galois correspondence, the unsolvable quintic, and impossible constructions
• Number Theory — Properties of integers, primality, and modular arithmetic
• Category Theory — The "math of math"; high-level abstractions used in functional programming
• Tensor Theory — Multilinear algebra and its applications from physics to LLM compression

Analysis

The rigorous theory beneath calculus.

• Real Analysis — Completeness, compactness, and the axiomatic foundation of calculus
• Complex Analysis — Calculus extended to complex numbers; residues and conformal maps
• Measure Theory — Generalizing "size" and "volume"; the foundation of modern probability
• Functional Analysis — Infinite-dimensional vector spaces and their operators
• Chaos and Dynamical Systems — Nonlinear evolution, strange attractors, and the limits of predictability

Geometry and Topology

Shape, space, and properties preserved under continuous deformation.

• Differential Geometry — Calculus on curved spaces and manifolds; the math of general relativity
• Topology (Mathematics) — Topological spaces, algebraic invariants, and persistent homology
• Topology — The conceptual view: continuous deformation and the shape of data

Probability and Statistics

Reasoning about uncertainty and analyzing data.

• Random Variables and Probability Distributions — The distribution zoo, expectation, LLN, and the Central Limit Theorem
• Probability Theory — The measure-theoretic foundations
• Bayesian Reasoning and Bayesian Inference — Updating beliefs; posterior computation in practice
• Markov Chain Fundamentals — Memoryless stochastic processes, MCMC, and PageRank
• Statistics Fundamentals — Descriptive statistics and data-as-vectors geometry
• Statistical Inference — Hypothesis testing, p-values, and power
• Regression Analysis — OLS as projection, Gauss-Markov, and ridge

Optimization and the Applied Toolkit

Topics with direct applications in algorithms, optimization, and system design.

• Convex Analysis and Optimization — Convexity, duality, KKT, and the tractable-problem hierarchy
• Optimization Algorithms — Gradient descent, Adam, L-BFGS, and the engines of ML training
• Linear Programming Foundations — Optimizing linear objectives subject to linear constraints
• Integer and Combinatorial Optimization — Finding the best solution in a discrete search space
• Numerical Methods — Solving continuous math problems in discrete computer arithmetic
• Combinatorics Refresher — Counting, permutations, and generating functions
• Graph Theory Deep Dive and Spectral Graph Theory — Networks and their spectra
• Game Theory Fundamentals — Strategic decision-making in competitive environments
• Information Theory — Entropy, mutual information, and the limits of communication
• Discrete Mathematics and Discrete Math Refresher — The digital spine
• Fuzzy Logic — Reasoning with degrees of truth rather than binary true/false
• Applied Math Survey — A high-level map of the mathematical tools in science and engineering

Advanced and Specialty Topics

• Information Geometry — Probability distributions as a Riemannian manifold
• Tropical Geometry — Min-plus algebra for scheduling and shortest paths
• Topos Theory and Higher Category Theory — Generalized universes and formal verification
• Algebraic Statistics and Singular Learning Theory — Loss landscapes through algebraic geometry
• Non-Standard Analysis and Ergodic Theory — Infinitesimals and time averages in finance
• Distributionally Robust Optimization — Optimizing against worst-case distributions

Adjacent Hubs

• ML Hub — The application of these mathematical tools to machine learning

• Operations Research Hub — Optimization and decision science in practice

• Quantitative Finance Research Hub — Stochastic and statistical machinery applied to markets

• Software Engineering Practices Hub — Applying rigorous logic to code and system design

• MonteCarloMethods: Statistical simulation, MCMC, and variance reduction techniques.

Operations Research and Stochastic Mathematics

• MarkovDecisionProcesses — Bellman optimality, Value/Policy iteration, and Q-learning reinforcement dynamics.
• LinearProgrammingSimplex — Standard form, Simplex tableau pivoting, and Strong Duality theorem.
• LagrangianMultipliers — Constrained optimization, Karush-Kuhn-Tucker (KKT) conditions, and shadow prices.
• ConvexOptimization — Convex sets, epigraphs, Slater condition, and ADMM decomposition.
• StochasticProcesses — Martingale theory, Poisson jump processes, and Ito stochastic calculus.
• BlackScholesModel — Geometric Brownian motion SDE, PDE derivation, and option Greeks.

Numerical Analysis & Mathematical Foundations

• NumericalDifferentialEquations — Runge-Kutta 4th-order (RK4), stiff ODE solvers, and Butcher tableaus.
• NumericalErrorAndStability — IEEE 754 floating point round-off, condition numbers, and backward stability.
• NumericalLinearAlgebra — QR factorization, Householder reflections, SVD decomposition, and GMRES.
• NumericalRootFinding — Newton-Raphson quadratic convergence, Brent's method, and polynomial deflation.
• AUniqueDiscovery — Foundational reflections on mathematical beauty, symmetries, and invariants.

Specialized Mathematical Fields

• [ChaosDynamical Hub](ChaosDynamical Hub) — Lorenz attractors, bifurcation theory, Lyapunov exponents, and nonlinear dynamics.
• [PredicateLogic Hub](PredicateLogic Hub) — First-order logic, quantifiers, Gödel completeness, and automated theorem proving.