# Lists of mathematics topics This page is a directory of directories: the map by which [[Mathematics]] organizes its own sprawl. Wikipedia maintains it as a list of lists, and the vault mirrors that function — but a good index is not a neutral object. How the discipline files itself (foundations, [[Abstract_algebra|algebra]], [[Mathematical_analysis|analysis]], [[Geometry|geometry]], [[Discrete_mathematics|discrete mathematics]], [[Probability_theory|probability]], applications) encodes real claims about what is fundamental to what, and the filing system has load-bearing consequences: it decides what a student meets first, how journals referee, and which subjects get treated as "core." The professional version is the Mathematics Subject Classification, maintained jointly by *Mathematical Reviews* and *zbMATH* and revised roughly each decade (MSC2020 is current), with about 63 two-digit top-level classes refining to over 5,000 leaf codes. What follows makes the taxonomy itself the technical content: the major divisions, why the boundaries sit where they sit, and where each division's anchor articles live in this vault. ## Foundations: the language layer Everything else is written in this layer's notation. [[Logic|Logic]] supplies the rules of inference, [[First-order_logic|first-order logic]] the standard formal grammar, and [[Formal_system|formal systems]] the general frame in which a [[Proposition|proposition]] is derivable or not. [[Set_theory|Set theory]] has served as the default ontology since the early twentieth century — every object a set, every function a set of pairs — while [[Category_theory|category theory]] (1945 onward) offers the rival, relation-first foundation. The boundary fact worth knowing: foundations is the one division whose theorems (Gödel, 1931) constrain the whole enterprise, including the [[Theory_of_computation|theory of computation]] that grew directly out of it, and its habits of [[Deductive_reasoning|deductive reasoning]] are what make the rest of the map trustworthy. ## The continuous and the discrete columns The deepest fold in the map separates the continuum from the countable. The continuous column descends from [[Calculus|calculus]] through [[Mathematical_analysis|analysis]] — [[Complex_analysis|complex analysis]], [[Functional_analysis|functional analysis]], [[Harmonic_analysis|harmonic analysis]], the [[Calculus_of_variations|calculus of variations]] — into the theory of the [[Differential_equation|differential equation]], split between the [[Ordinary_differential_equation|ordinary]] and [[Partial_differential_equation|partial]] kinds, with [[Integral_transform|integral transforms]] as the connecting tissue. The discrete column runs through [[Combinatorics|combinatorics]], [[Graph_theory|graph theory]], number theory, and [[Algorithm|algorithmics]] toward computation. The fold is real but porous: [[Discrete_time_and_continuous_time|discrete and continuous time]] model the same dynamics, generating functions solve counting problems with analysis, and analytic number theory proves discrete facts with contour integrals. Most modern applied work — [[Computational_mathematics|computational mathematics]] above all — consists precisely of moving problems across this fold, discretizing the continuous so a machine can touch it. ## Space and structure: geometry, topology, algebra The geometric division classifies by how much structure a space carries. [[Geometry|Geometry]] proper spans the metric classics; [[Differential_geometry|differential geometry]] adds smoothness and curvature on [[Manifold|manifolds]]; [[Algebraic_geometry|algebraic geometry]] studies zero sets of polynomials; [[Projective_geometry|projective]], [[Affine_geometry|affine]], [[Finite_geometry|finite]], [[Discrete_geometry|discrete]], and [[Computational_geometry|computational geometry]] each keep a different subset of the axioms. [[Topology|Topology]] keeps almost none — only continuity — and is correspondingly the most general, with [[General_topology|general topology]] at the point-set base and invariants like the [[Euler_characteristic|Euler characteristic]] and [[Genus_(mathematics)|genus]] classifying [[Surface_(topology)|surfaces]]. The structural division, algebra, classifies by operations instead of spaces: [[Group_theory|group theory]] for symmetry, [[Field_(mathematics)|fields]] for arithmetic, [[Linear_algebra|linear algebra]] for vector spaces and the [[Tensor|tensors]] built on them, [[Lie_group|Lie groups]] where algebra and geometry fuse. That these divisions interlock — every geometry has a symmetry group, per Klein's 1872 Erlangen program — is the strongest argument that the map describes one subject, not many. ## The applied belt, keyed to this vault The outer belt is where mathematics faces the world, and it is the belt this vault inhabits most densely. The table pairs each applied district with its vault anchors. | District | What it optimizes or explains | Vault anchors | |---|---|---| | Chance and data | inference under [[Uncertainty|uncertainty]] | [[Probability_theory|Probability theory]], [[Statistics|Statistics]], [[Time_series|Time series]], [[Monte_Carlo_method|Monte Carlo method]] | | Decision and strategy | choice among agents | [[Game_theory|Game theory]], [[Decision_theory|Decision theory]], [[Operations_research|Operations research]], [[Mathematical_optimization|Mathematical optimization]] | | Signals and control | steering and sensing | [[Control_theory|Control theory]], [[Information_theory|Information theory]], [[Signal_processing|Signal processing]], [[Kalman_filter|Kalman filter]] | | Dynamics | change over time | [[Dynamical_system|Dynamical system]], [[Chaos_theory|Chaos theory]], [[Nonlinear_system|Nonlinear system]], [[Phase_space|Phase space]] | | Computation | effective procedure | [[Theory_of_computation|Theory of computation]], [[Computational_mathematics|Computational mathematics]], [[Computer_algebra|Computer algebra]], [[Cryptography|Cryptography]] | | Science interfaces | domain modeling | [[Mathematical_physics|Mathematical physics]], [[Mathematical_chemistry|Mathematical chemistry]], [[Mathematical_and_theoretical_biology|Mathematical biology]], [[Econometrics|Econometrics]], [[Engineering_mathematics|Engineering mathematics]] | ## How to use a map that is also an argument Three honest caveats. First, boundaries drift: probability was analysis until Kolmogorov's 1933 axioms gave it its own district; computation did not exist as a division before the 1930s. Second, list pages flatten hierarchy — the vault's sibling inventories ([[List_of_numerical_libraries|numerical libraries]], [[Lists_of_engineering_software|engineering software]], [[List_of_systems_scientists|systems scientists]]) show the same trade of depth for coverage, and a [[Body_of_knowledge|body of knowledge]] always looks tidier in the index than in practice. Third, proximity in the file system is not proximity in method: [[Statistical_mechanics|statistical mechanics]] sits in physics yet shares its combinatorial core with [[Entropy_(information_theory)|information-theoretic entropy]]. Read this page, in short, the way a systems reader reads any [[Hierarchy|hierarchy]] imposed on a [[Complex_system|complex system]] — as one useful projection of a [[Network_theory|network]] that is, in truth, densely and gloriously cross-linked. **On the spine:** [[Mathematics]] · [[Applied_mathematics]] · [[Discrete_mathematics]] · [[Mathematical_analysis]] · [[Geometry]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Lists_of_mathematics_topics) : [Wikitube](https://en.wikitube.io/wiki/Lists_of_mathematics_topics) ## Previous hub tags Hubs: `Systems`. Portals: [[PORTAL_Graph_theory]], [[PORTAL_Decision_theory]], [[PORTAL_Information_theory]], [[PORTAL_Control_theory]], [[PORTAL_Operations_research]]. --- *Repopulated 2026-08-12 · redlink fill · 0 deletions.*