A Bird’s-Eye View of Mathematics

When you try to organize the fields of mathematics as knowledge, big buckets like “algebra” and “analysis” come to mind immediately, but how many finer fields sit inside them, and how they connect to each other, is surprisingly hard to see.

This article organizes what fields exist in mathematics, flatly and comprehensively. It covers not just the top-level classification but one level deeper, down to subfields you can survey at a glance. It doesn’t explain individual theorems or theories themselves; it sticks to classification as a map.

Conclusions

• For classifying mathematical fields there is an industry-standard classification system, the Mathematics Subject Classification (MSC). Journals and arXiv, zbMATH, MathSciNet, and others actually use it
• MSC has a tree structure branching from roughly 2-digit field codes (about 60 of them) at the top down into finer 3-digit and 5-digit subcodes
• Bundled loosely to match how they feel in learning and use, they organize readily into about 8 groups: “foundations and logic,” “algebra,” “number theory and discrete mathematics,” “analysis,” “geometry and topology,” “probability and statistics,” “applied and computational mathematics,” and “interdisciplinary areas”
• Each of the 8 groups has another level of finer subfields inside it. For instance, algebra contains group theory, ring theory, representation theory, and so on, and the subfields also serve as foundations for one another
• The classifications and code correspondences shown here are a rough organization for the sake of an overview; for exact notation and the latest version you should always check the primary sources (see the reference links below)

Prerequisites

• Intended reader: someone who wants a reasonably comprehensive grasp of the overall landscape of mathematical fields
• Aim: not to explain individual theories, but to build a flat map of “what fields exist and how they are divided”
• Classification system referenced: MSC2020 (Mathematics Subject Classification, 2020 edition) as the main axis
• Note that it doesn’t necessarily match the curricular divisions of mathematics education (departmental structures at universities, etc.)

How to Divide Up the Fields of Mathematics

Dividing mathematics into fields varies with historical circumstance and the customs of research communities, so there is no single absolute right answer. That said, if you want to reference fields across the board and reproducibly, the following classification systems are the de facto standard in practice.

• MSC (Mathematics Subject Classification): a classification system maintained jointly by the American Mathematical Society (AMS) and zbMATH Open. Widely used as field codes when submitting papers, and revised every few years (most recently MSC2020)
• arXiv’s math categories: the classification used when submitting preprints. Not as fine-grained as MSC, but good for seeing the distribution of contemporary research
• Wikipedia’s Areas of mathematics: easy to consult as a general-audience overview, but not an official classification system

MSC starts from 2-digit top-level categories (for example, 11 is number theory and 53 is differential geometry) and branches into finer 3-digit and 5-digit field codes. Counts and exact notation change from version to version, so this article avoids reproducing a strict code listing and limits itself to introducing representative top-level categories and where they sit. For an accurate listing, please check the official MSC2020 distribution (see the reference links below).

An Overview Map by Major Field

Bundling MSC’s top-level categories loosely into 8 groups, matching how they’re used in learning and in practice, gives the following. This is a bundling for the sake of an overview only, not MSC’s own official grouping.

flowchart TB
  math[Mathematics] --> foundation[Foundations & Logic]
  math --> algebra[Algebra]
  math --> numtheory[Number Theory & Discrete Math]
  math --> analysis[Analysis]
  math --> geometry[Geometry & Topology]
  math --> prob[Probability & Statistics]
  math --> applied[Applied & Computational Math]
  math --> interdisciplinary[Interdisciplinary Areas]

The arrows represent the “containment relation” from mathematics as a whole to each top-level category.

Representative example fields for each group are as follows.

The codes are approximate, and some fields span several top-level categories. For example, representation theory relates to both algebra (20) and analysis (22, 43), and statistical mechanics can be discussed from the perspective of analysis, probability, or physics.

Digging Into the Subfields of Each Group

Top-level categories alone are too coarse, so let’s go one level deeper into representative subfields for each of the 8 groups.

Axiom Systems

Each field of mathematics rests on “axioms,” premises accepted as true without proof. Even for the same objects, changing the axioms you adopt yields a different theory, so organizing the representative axiom systems first makes it easier to understand where each field sits.

Axiom systems are not independent of one another; they often have a hierarchical relationship, such as the Peano axioms and the axioms of an ordered field being constructible on top of ZFC. On the other hand, there are cases like Euclid’s axioms where replacing part of the axioms gives rise to a different theoretical system.

A Closer Look at the ZFC Axioms

ZFC lays down as 9 axioms the properties that sets themselves must satisfy.

1. Axiom of extensionality: sets with the same elements are equal
2. Axiom of the empty set: there exists a set with no elements (the empty set)
3. Axiom of pairing: for any two sets a, b there exists a set whose only elements are a and b
4. Axiom of union: given a set of sets, there exists the union collecting all of their elements
5. Axiom of the power set: for any set there exists the set of all its subsets (the power set)
6. Axiom of infinity: there exists a set with infinitely many elements, corresponding to the natural numbers
7. Axiom schema of replacement: when each element of a set is assigned another element by a functional correspondence, the image is again a set
8. Axiom of regularity (foundation): every non-empty set has an element that shares no element with the set itself. This rules out self-referential sets such as $A \in A$
9. Axiom of choice: for any family of non-empty sets there exists a function (a choice function) picking one element from each set

Items 1–8 constitute the system ZF, which “does not include the axiom of choice”; adding item 9, the axiom of choice, gives ZFC. Not only set theory itself, but most objects of modern mathematics — natural numbers, reals, functions, topological spaces — can ultimately be constructed as sets within the ZFC framework.

Of these 9 axioms, the ones that unconditionally assert “a set exists out of nothing” are effectively limited to just two: the axiom of the empty set and the axiom of infinity. Pairing, union, power set, the replacement schema, and regularity are all conditional assertions of the form “if some set already exists, then such a new set also exists” — they merely assemble new sets out of existing ones as material. In fact, many textbooks don’t posit the axiom of the empty set independently, instead extracting the empty set from the set guaranteed by the axiom of infinity via the separation schema (as in $\{x \in X : x \neq x\}$); in that case the axiom of infinity alone carries the primary assertion that “a set exists.”

A Closer Look at the Axiom of Choice

The axiom of choice differs in character from the other 8 axioms. It merely asserts that “a choice function exists” without showing a way to construct that function concretely, so it is called a non-constructive axiom.

• Independence: the axiom of choice can be neither proved nor refuted from the other axioms of ZF. Gödel showed that “if ZF is consistent then so is ZFC” (1938), and Cohen showed that “if ZF is consistent then so is the system with the negation of the axiom of choice added” (1963). In other words, whether to adopt the axiom of choice is an independent choice that cannot be decided from the other axioms
• Equivalent propositions: under ZF, the axiom of choice is logically equivalent to the following propositions
  • Zorn’s lemma: a partially ordered set in which every chain has an upper bound has a maximal element
  • Well-ordering theorem: any set can be given a well-ordering
  • Tychonoff’s theorem: the product of any number of compact spaces is compact
• Applications: the axiom of choice is used throughout analysis and algebra, for example in the existence of a basis for any vector space and in the construction of non-measurable sets of reals
• Controversial consequences: counterintuitive results such as the Banach-Tarski paradox (splitting a ball into finitely many pieces and reassembling them into two balls of the same size as the original) also follow from the axiom of choice. For this reason, there are styles of mathematics that avoid the axiom of choice, or that make its use explicit

Relationship with Euclid’s Axioms

There is a structurally similar relationship between the axiom of choice and geometry’s Euclidean axioms (particularly the parallel postulate).

• Independence as a common feature: for over 2000 years many mathematicians tried to prove that the parallel postulate should follow from the other 4 postulates, and failed. In the 19th century, Lobachevsky, Bolyai, Gauss, and others constructed consistent systems in which the parallel postulate is negated (non-Euclidean geometry), showing that the parallel postulate is independent of the other postulates. This has the same structure as the axiom of choice being independent of the other axioms of ZF
• Re-choosing axioms gives a different theory: just as adopting the parallel postulate yields Euclidean geometry while negating it yields hyperbolic or elliptic geometry, adopting the axiom of choice yields ordinary set theory (ZFC) while negating it yields a set theory in which the axiom of choice fails (ZF + ¬AC). Both are internally consistent theoretical systems
• A difference in subject matter: whereas Euclid’s axioms are an axiom system defining concrete objects — “space and figures” — ZFC is an axiom system that serves as the foundation for constructing nearly every object in mathematics, namely “sets.” Euclid’s axioms are the framework of a single field, geometry, while ZFC provides the framework for mathematics as a whole; their reach differs

Foundations & Logic

flowchart TB
  foundation[Foundations & Logic] --> logic[Mathematical Logic]
  foundation --> settheory[Set Theory]
  foundation --> modeltheory[Model Theory]
  foundation --> prooftheory[Proof Theory]
  foundation --> category[Category Theory]
  modeltheory --> logic
  modeltheory --> settheory
  prooftheory --> logic

Arrows basically represent “field → subfield contained under it.” However, the 3 arrows Model Theory → Mathematical Logic, Model Theory → Set Theory, and Proof Theory → Mathematical Logic represent a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Mathematical logic: formally treats the structure and properties of logic itself, such as propositional and predicate logic
• Set theory: treats properties of sets that form the foundation of mathematics, such as cardinality, ordinals, and the axiom of choice
• Model theory: studies how a logical formula is interpreted within a concrete mathematical structure. Built on mathematical logic and set theory
• Proof theory: analyzes the structure and strength of proofs themselves as mathematical objects. Built on mathematical logic
• Category theory: focuses on the relationship between objects and morphisms (maps), describing the structures of different fields uniformly

Algebra

flowchart TB
  algebra[Algebra] --> linear[Linear Algebra]
  algebra --> group[Group Theory]
  algebra --> ring[Ring Theory]
  algebra --> field[Field Theory]
  algebra --> module[Module Theory]
  algebra --> rep[Representation Theory]
  algebra --> homological[Homological Algebra]
  rep --> group
  rep --> ring
  homological --> module

Arrows basically represent “field → subfield contained under it.” However, the 3 arrows Representation Theory → Group Theory, Representation Theory → Ring Theory, and Homological Algebra → Module Theory differ in meaning, representing a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Linear algebra: treats vector spaces and linear maps; a basic field used as a foundation by other fields too
• Group theory: captures symmetry through the structure called a “group”
• Ring theory and field theory: study the properties of algebraic structures (rings, fields) that have addition and multiplication
• Module theory: generalizes the idea of vector spaces over a ring
• Representation theory: “represents” the elements of groups and rings as linear maps and investigates them with the tools of linear algebra. Built on group theory and ring theory
• Homological algebra: captures relationships between modules via chain complexes and the like, connecting also to topology and algebraic geometry. Built on module theory

Number Theory & Discrete Mathematics

flowchart TB
  numtheory[Number Theory & Discrete Math] --> elementary[Elementary Number Theory]
  numtheory --> algebraicnum[Algebraic Number Theory]
  numtheory --> analyticnum[Analytic Number Theory]
  numtheory --> combinatorics[Combinatorics]
  numtheory --> graphtheory[Graph Theory]
  numtheory --> order[Order & Lattice Theory]
  algebraicnum --> elementary
  analyticnum --> elementary
  graphtheory --> combinatorics

Arrows basically represent “field → subfield contained under it.” However, the 3 arrows Algebraic Number Theory → Elementary Number Theory, Analytic Number Theory → Elementary Number Theory, and Graph Theory → Combinatorics represent a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Elementary number theory: treats properties of the integers themselves, such as primes and congruences
• Algebraic number theory: investigates number-theoretic properties within algebraic extension fields. Built on elementary number theory
• Analytic number theory: treats number-theoretic objects with the methods of analysis, as with the distribution of primes. Built on elementary number theory
• Combinatorics: treats counting and structure of finite sets
• Graph theory: studies network structures made of vertices and edges. Built on combinatorics
• Order & lattice theory: treats structures such as ordered sets and lattices

Analysis

flowchart TB
  analysis[Analysis] --> calculus[Calculus]
  analysis --> real[Real Analysis]
  analysis --> complex[Complex Analysis]
  analysis --> functional[Functional Analysis]
  analysis --> harmonic[Harmonic Analysis]
  analysis --> diffeq[ODEs & PDEs]
  analysis --> dynamical[Dynamical Systems]
  real --> calculus
  complex --> calculus
  functional --> real
  harmonic --> functional
  diffeq --> calculus
  dynamical --> diffeq

Arrows basically represent “field → subfield contained under it.” However, the 6 arrows from Real Analysis → Calculus onward represent a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Calculus: treats rates of change and accumulation of functions; the entrance to analysis
• Real analysis: rigorously treats properties of functions and limits on the reals. Includes measure theory. Built on calculus
• Complex analysis: treats properties of holomorphic functions defined over the complex numbers. Built on calculus
• Functional analysis: treats infinite-dimensional vector spaces such as Banach and Hilbert spaces. Built on real analysis
• Harmonic analysis: treats Fourier analysis and its generalizations. Built on functional analysis
• ODEs & PDEs: the equations describing change of functions become the object of study themselves. Built on calculus
• Dynamical systems: studies the long-term, qualitative behavior of time-evolving systems. Built on differential equations

Geometry & Topology

flowchart TB
  geometry[Geometry & Topology] --> euclidean[Euclidean Geometry]
  geometry --> differential[Differential Geometry]
  geometry --> algebraicgeo[Algebraic Geometry]
  geometry --> pointset[Point-Set Topology]
  geometry --> algebraictopo[Algebraic Topology]
  geometry --> convex[Convex Geometry]
  geometry --> manifold[Manifold Theory]
  manifold --> pointset
  differential --> manifold
  algebraictopo --> pointset
  convex --> euclidean

Arrows basically represent “field → subfield contained under it.” However, the 4 arrows Manifold Theory → Point-Set Topology, Differential Geometry → Manifold Theory, Algebraic Topology → Point-Set Topology, and Convex Geometry → Euclidean Geometry represent a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Euclidean geometry: treats properties of classical figures such as lines, circles, and polygons
• Differential geometry: treats properties of curves, surfaces, and manifolds with the tools of calculus. Built on manifold theory
• Algebraic geometry: captures figures as zero sets of polynomials and investigates them with the tools of algebra
• Point-set topology: treats basic properties of topological spaces, such as open sets and continuity
• Algebraic topology: investigates properties of topological spaces with algebraic tools such as homology groups. Built on point-set topology
• Convex geometry: treats properties of convex sets. Built on Euclidean geometry
• Manifold theory: treats properties of spaces that locally look like Euclidean space (manifolds). Built on point-set topology

Probability & Statistics

flowchart TB
  prob[Probability & Statistics] --> probability[Probability Theory]
  prob --> stochastic[Stochastic Processes]
  prob --> statistics[Statistics]
  prob --> infotheory[Information Theory]
  stochastic --> probability
  statistics --> probability
  infotheory --> probability

Arrows basically represent “field → subfield contained under it.” However, the 3 arrows Stochastic Processes → Probability Theory, Statistics → Probability Theory, and Information Theory → Probability Theory represent a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Probability theory: treats uncertainty mathematically, via random variables and probability distributions
• Stochastic processes: treats probabilistic phenomena that change over time. Built on probability theory
• Statistics: the field concerned with methods for inferring properties of a population from data. Built on probability theory
• Information theory: treats transmission and compression of information mathematically, via quantities such as information content and entropy. Built on probability theory

Applied & Computational Mathematics

flowchart TB
  applied[Applied & Computational Math] --> numerical[Numerical Analysis]
  applied --> optimization[Optimization & Operations Research]
  applied --> control[Control Theory]
  applied --> game[Game Theory]
  applied --> cstheory[Mathematical Foundations of CS]
  control --> optimization

Arrows basically represent “field → subfield contained under it.” However, the single arrow Control Theory → Optimization & Operations Research represents a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Numerical analysis: treats methods for numerically approximating equations and functions
• Optimization & operations research: treats methods for maximizing and minimizing objective functions
• Control theory: designing inputs to drive a system to a desired state. Built on optimization
• Game theory: treats optimal strategies in situations involving multiple decision makers
• Mathematical foundations of computer science: treats computation itself mathematically, via complexity theory, automata theory, and so on

Interdisciplinary Areas

flowchart TB
  interdisciplinary[Interdisciplinary Areas] --> mathphysics[Mathematical Physics]
  interdisciplinary --> mathbio[Mathematical Biology]
  interdisciplinary --> matheconomics[Mathematical Economics]
  interdisciplinary --> statmech[Statistical Mechanics]
  interdisciplinary --> relativity[Relativity]
  statmech --> mathphysics
  relativity --> mathphysics

Arrows basically represent “field → subfield contained under it.” However, the 2 arrows Statistical Mechanics → Mathematical Physics and Relativity → Mathematical Physics represent a dependency relation, “subfield → another subfield it uses as a theoretical foundation.”

• Mathematical physics: studies the mathematical formulation of physical phenomena and its structure
• Mathematical biology: describes and analyzes biological phenomena with mathematical models
• Mathematical economics: describes and analyzes economic phenomena with mathematical models
• Statistical mechanics: treats properties of systems made of many particles probabilistically. Positioned as a subfield of mathematical physics
• Relativity: treats the structure of spacetime geometrically. Positioned as a subfield of mathematical physics

As this shows, subfields don’t merely hang beneath a top-level category; many, like representation theory and homological algebra in algebra, come into existence only because other subfields do.

Relationships Across Fields

So far we’ve looked at the internal structure of fields. In actual mathematics, tools and viewpoints frequently cross field boundaries. Finally, let’s organize cross-field relationships from both concrete examples and a macro perspective.

Cross-Field Connections Seen Through Concrete Examples

• Differentiation in analysis can be expressed with matrices from linear algebra (the Jacobian, for instance). The derivative of a multivariable function is handled as a linear map with the tools of algebra
• Rotation operations via matrices in linear algebra represent rotation itself in geometry. Matrices, objects of algebra, describe geometric transformations
• The stability of a differential equation (analysis) can be investigated using the eigenvalues of a matrix (algebra) as a clue
• Analyzing stochastic processes (probability & statistics) uses measure theory (analysis) and operator theory (functional analysis)
• The adjacency matrix in graph theory (number theory & discrete math) can be analyzed with the tools of linear algebra

Cross-Field Relationships from a Macro Perspective

Beyond individual cases, surveying macroscopically how fields supply tools to one another gives the following.

flowchart LR
  foundation[Foundations & Logic] -.-> algebra[Algebra]
  algebra -.-> analysis[Analysis]
  algebra -.-> geometry[Geometry & Topology]
  algebra -.-> numtheory[Number Theory & Discrete Math]
  analysis -.-> geometry
  analysis -.-> prob[Probability & Statistics]
  analysis -.-> applied[Applied & Computational Math]
  prob -.-> applied
  geometry -.-> interdisciplinary[Interdisciplinary Areas]
  analysis -.-> interdisciplinary
  prob -.-> interdisciplinary

The dashed arrows differ in meaning from the earlier diagrams. They represent an application relationship: describing or analyzing one field with the tools or viewpoint of another. Please read them as distinct from the containment and dependency relations shown by solid lines.

• Algebra is a common language used across fields. In the vocabulary of vectors and matrices from linear algebra, you can describe differential operators in analysis, rotations and transformations in geometry, graph structures in number theory & discrete math, and more
• Analysis, as a framework for treating continuous change, supplies tools to differential geometry in geometry, measure theory in probability & statistics, and numerical analysis in applied & computational math
• Geometry and analysis connect through differential geometry and manifold theory, becoming the theoretical foundation for interdisciplinary theories such as relativity
• Probability & statistics builds on measure theory from analysis while also extending into applied and interdisciplinary areas, as with information theory and statistical mechanics
• Category theory in foundations & logic serves as a viewpoint that describes the structures of algebra, geometry, and analysis uniformly. It’s frequently used across fields
• Interdisciplinary areas are not closed to the tools of a single field. They handle real-world phenomena with tools from multiple fields — algebra, analysis, geometry, probability, and so on

In this way, cross-field relationships are at once a collection of individual cases and a larger cycle in which algebra, analysis, and geometry supply tools to one another.

Caveats

• MSC’s classification reflects historical circumstance and the customs of research communities, and doesn’t necessarily coincide with “the mathematically most natural classification”
• Field boundaries are fluid, and new fields that existing classification codes alone can’t fully capture — such as the theoretical foundations of data science — are emerging
• The top-level categories and code correspondences given in this article are a simplified version for overview purposes only; for exact notation and the latest version you should check the primary source (the official MSC2020 distribution)
• University departmental structures (applied mathematics, information science, and so on) are a separate, practical division from MSC’s classification and don’t necessarily map onto it directly

References

• MSC2020 - Mathematics Subject Classification (AMS)
• zbMATH Open - Mathematics Subject Classification
• arXiv - Mathematics category taxonomy
• Wikipedia - Areas of mathematics