You study this in Grade 8–12. It is the foundation.
Introduced in Grade 12 or early university. The gateway.
Accessible once you have the foundation. Not out of reach.
14 territories. Click any node to explore.
Each territory on the map is a major branch of mathematics. The dashed lines show which branches feed into each other. The orange nodes are in your CAPS curriculum. Indigo nodes are partially there. The rest is the rest of mathematics — waiting.
Not in SA school curriculum — the deep end of the pool.
Introduced at Grade 12 or first-year university — the transition zone.
Fully covered in Grade 8–12. The foundation for everything above.
Every territory in full detail.
Each branch below: what it studies, why it matters, the most famous open problem, and its direct connection to South African education and careers.
Foundations
The bedrock underneath everything else.
Logic, Set Theory, Proof Theory, Model Theory — these are the meta-questions of mathematics. What is a proof? What can be computed? What is a number? The foundations were formalised in the 19th–20th century and are still actively disputed.
Whether the Continuum Hypothesis is provably true or false in any consistent axiomatic system (Gödel / Cohen, 1963: it is independent of ZFC).
Underpins computer science, AI verification, and formal methods in software engineering — all growing fields in SA tech.
Arithmetic & Number Theory
The oldest and deepest question: what are numbers?
Number Theory begins with the integers and asks questions that are deceptively simple and fiendishly hard. Fermat's Last Theorem looked elementary; it took 358 years and 129 pages to prove. The Riemann Hypothesis has been open for 166 years.
The Riemann Hypothesis: do all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2? Millennium Prize: $1,000,000.
Cryptographic Number Theory underpins banking, SARS systems, and every HTTPS connection — directly relevant to SA fintech.
Algebra
The grammar of mathematical structure.
School algebra (solving equations) is the surface. Abstract Algebra studies the structure underneath: groups, rings, fields, and the symmetries that govern them. Emmy Noether's theorem connecting symmetry to conservation laws may be the most important result in 20th-century physics.
The Langlands Program: a grand unified theory connecting number theory, representation theory, and automorphic forms. One of the most ambitious open projects in mathematics.
Group Theory → cryptography → banking. Representation Theory → quantum computing, growing at UCT and Wits.
Geometry
The mathematics of shape, space, and curvature.
Euclid's five postulates dominated for 2,000 years. When Riemann asked what happens if the fifth postulate is dropped, Non-Euclidean Geometry was born — and Einstein used it 60 years later for General Relativity. Geometry is both ancient and urgently modern.
The Geometrisation Conjecture (Thurston) — proved by Perelman in 2003 with Ricci flow. The closest landmark is now the Smooth Poincaré Conjecture in dimension 4.
Computational Geometry → GIS, satellite imagery, urban planning. Differential Geometry → physics modelling at SAAO.
Topology
The mathematics of continuity and connectivity.
Topology asks: what properties of a shape survive any continuous deformation? A donut and a coffee mug are topologically identical (both have one hole). Knot Theory has applications from DNA structure to quantum computing. Algebraic Topology connects discrete and continuous worlds.
The Poincaré Conjecture in dimension 4 (smooth case) remains open. Knot invariants and their connections to quantum field theory are actively studied.
Topological Data Analysis → medical imaging, genomics. Applied at UCT's Computational Biology group.
Analysis
The rigorous foundation of calculus.
Newton and Leibniz invented calculus without rigour. A century later, Cauchy, Riemann, and Weierstrass rebuilt it from scratch, asking: what exactly is a limit? Analysis is calculus made rigorous, then generalised far beyond anything in the school curriculum.
The Navier-Stokes Existence and Smoothness problem: do smooth solutions always exist for the equations governing fluid flow? Millennium Prize: $1,000,000.
Functional Analysis → quantum mechanics modelling at SA national labs. Harmonic Analysis → signal processing, used in SA telecoms.
Differential Equations
The language of change, written in calculus.
Everything that changes continuously — populations, temperatures, fluid pressure, electrical currents, epidemic spread — can be described by differential equations. PDEs govern weather, climate, and quantum particles. Stochastic DEs govern financial markets.
Navier-Stokes (fluid turbulence) and Yang-Mills existence/mass gap (quantum field theory) are both Millennium Problems related to differential equations.
Epidemiological modelling — UKZN and UCT used Stochastic DEs to model COVID-19 spread in SA during 2020–2022.
Probability & Statistics
The mathematics of uncertainty.
Probability gives uncertainty a language. Statistics gives data a voice. Together they underpin every scientific claim, every clinical trial, every actuarial table, every machine learning model, and every economic forecast — including the ones that are wrong.
Foundations of Bayesian inference and its relationship to frequentist statistics remain philosophically contested. Extreme Value Theory has major open questions for climate risk.
Direct CAPS relevance. SA actuarial profession is world-class. Stats SA, CSIR, and academic departments all hire graduate statisticians.
Discrete Mathematics
The mathematics of finite and countable structures.
Combinatorics counts. Graph Theory models. Coding Theory corrects errors in digital transmission. Every computer network, every communication protocol, every algorithm runs on the mathematics of discrete structures.
P vs NP: can every problem whose solution can be verified quickly also be solved quickly? The most famous open problem in computer science. Millennium Prize: $1,000,000.
Coding Theory → telecommunications (MTN, Vodacom). Graph Theory → social network analysis. CSIR uses combinatorial optimisation for logistics.
Applied Mathematics
Mathematics in the service of everything else.
Applied Mathematics uses the tools of pure mathematics to solve concrete problems: optimise supply chains, model financial markets, simulate fluid flow, design control systems. The boundary between "pure" and "applied" has never been clear — and that's fine.
Mathematical Biology: the mathematics of protein folding, network robustness, and evolutionary dynamics. Climate modelling: accurate multi-decadal prediction under uncertainty.
Mathematical Finance → JSE, actuarial practice. Operations Research → Transnet, Eskom, SAA scheduling. Climate modelling → SAWS.
Computational Mathematics
Making mathematics computable.
Computational Mathematics asks: how do we solve mathematical problems when exact solutions don't exist or can't be computed symbolically? Numerical Analysis, Scientific Computing, and Computer Algebra Systems are the tools that make modern engineering, climate science, and genomics possible.
Fast matrix multiplication: what is the true exponent ω of matrix multiplication? The current bound is ω < 2.37. Achieving ω = 2 is an open problem with major practical consequences.
CHPC (Centre for High Performance Computing) in Cape Town runs computational mathematics at scale for climate, genomics, and physics research.
Mathematical Logic & Computer Science
The mathematics that created computing.
Turing's model of computation, Gödel's incompleteness theorems, Shannon's information theory, and the P vs NP problem all emerged from asking: what are the limits of mathematics itself? This branch created the computer, the internet, and every piece of software ever written.
P vs NP (see Discrete Mathematics). Quantum Complexity: BQP vs PH — can quantum computers solve problems outside the polynomial hierarchy? Unknown.
Cryptography → SARS, banking, e-government. Formal Verification → safety-critical systems. Quantum Computing emerging at Wits and UCT.
Interdisciplinary Mathematics
Mathematics dissolving into everything.
Mathematics does not stay inside its own borders. Data Science, Machine Learning Theory, Network Science, Epidemiology, Neuroscience — all are now substantially mathematical. The fastest-growing mathematical fields in the 21st century are the ones that refused to stay in their lane.
The mathematical foundations of deep learning: why do overparameterised neural networks generalise? Why does gradient descent find good minima? Largely open.
ML Theory → AI development at Naspers, Standard Bank. Epidemiology → SAMRC, NICD. Network Science → telecoms optimisation.
History, Philosophy & Education
The conversation mathematics has with itself.
What is a mathematical proof? Are mathematical objects discovered or invented? Why does mathematics describe physical reality so precisely? These are not soft questions — they shape how mathematicians work, how mathematics is taught, and what counts as valid reasoning.
The Unreasonable Effectiveness of Mathematics (Wigner, 1960): why does pure mathematics developed for aesthetic reasons keep turning out to describe physical reality? No consensus answer.
Mathematics Education is a national priority field. AMESA (Association for Mathematics Education of SA) and multiple SA research chairs focus on this directly.
Mathematics is not 14 subjects. It is one — with 14 entry points.
Every branch feeds and is fed by others. Number Theory feeds Algebra feeds Topology feeds Analysis. Statistics feeds Machine Learning feeds back into everything. Here are the most important connections.
Algebraic Geometry — Grothendieck's revolution. Wiles proved Fermat's Last Theorem by connecting Number Theory → Algebra → Algebraic Geometry in one chain.
Analysis provides the rigorous foundations (limits, convergence, compactness) that make the study of differential equations meaningful rather than hand-wavy.
Every machine learning model is a statistical model. Every statistical model rests on probability theory. These three are not separate disciplines — they are one discipline in three registers.
Graph Theory, Number Theory, and Combinatorics together underpin every cryptographic protocol protecting every digital transaction on earth.
Algebraic Topology translates topological questions into algebraic ones (homotopy groups, homology). Knot Theory connects back to quantum field theory via the Jones polynomial.
Set Theory and Logic are the invisible scaffolding. Gödel's incompleteness theorems mean that even the foundations have limits — a profound and disturbing result that every mathematician lives with.
Every encryption protocol — RSA, elliptic-curve, TLS — is pure Number Theory applied. The most direct path from abstract mathematics to national security and daily banking.
Topology emerged from geometry by asking: what properties survive continuous deformation? Riemann's surface theory and Poincaré's analysis situs are the same subject seen from different angles.
Mathematical Optimisation, Stochastic Processes, and Differential Equations are the theoretical backbone of machine learning. Applied Mathematics and Data Science are converging fields.
The part that matters right now
is the part your grade is in.
Start with the 8%. Master it completely. Then the 92% opens in front of you, one branch at a time.