A structured visual guide to the major mathematical areas and their relationships.
This page presents a hierarchical view of mathematics using the MSC2020 structure. It highlights the broad branches, the main topic classes, and the finer subtopics that organize modern mathematics.
Search by code, branch, topic, subtopic, or a keyword from the descriptions.
This part of mathematics collects the broad conceptual tools that frame the discipline: foundations, history, logic, and the study of mathematical language itself.
Broad reference material, expository writing, and the organization of mathematical knowledge at a high level.
Studies the development of mathematics across cultures and periods, including the lives of influential mathematicians and the historical context of major discoveries.
Investigates the formal rules governing mathematical reasoning, proof, and the limits of what can be proved.
This part of mathematics studies discrete patterns, algebraic structures, and the combinatorial relationships that arise in finite or countable systems.
Concerned with counting, arranging, and structuring finite objects, often revealing patterns, symmetries, and extremal behavior.
Studies ordering relations and algebraic systems in which comparison and structure interact.
Provides a unifying framework for operations, identities, and general algebraic structures.
Studies the arithmetic properties of integers, primes, congruences, and Diophantine equations.
Focuses on fields, polynomial equations, and algebraic extensions.
Studies commutative rings, ideals, modules, and their geometric significance.
Studies geometric objects defined by polynomial equations and the links between algebra and geometry.
Examines vector spaces, matrices, eigenvalues, and linear transformations.
Generalizes familiar algebraic systems such as matrices and polynomial rings.
Studies algebraic systems where the associative law may fail, including Lie and Jordan algebras.
Formalizes structural patterns across mathematics and studies algebraic invariants through complexes and derived functors.
Assigns algebraic invariants to spaces and rings to capture subtle structural information.
Studies symmetry through groups, permutations, and related algebraic structures.
Combines algebra with analysis by studying groups that also carry topological or differentiable structure.
This branch examines the behavior of functions, limits, continuity, and the mathematics of change and accumulation.
Studies real-valued functions, limits, continuity, differentiation, and integration.
Formalizes notions of size, probability, and integration in general settings.
Explores holomorphic functions, conformal maps, and the rich geometry of complex analysis.
Studies harmonic functions, capacity, and the behavior of fields generated by sources and boundaries.
Extends complex analysis to higher-dimensional settings and analytic spaces.
Investigates functions that arise naturally in analysis, geometry, physics, and differential equations.
Studies equations describing how quantities evolve with respect to a single variable.
Studies equations involving several variables and their use in modeling physical and geometric systems.
Studies long-term behavior under iteration and evolution over time.
Studies equations involving shifts, recurrences, and functional identities that characterize families of functions.
Examines convergence, divergence, and summability methods for infinite sequences and series.
Analyzes functions via Fourier methods, singular integrals, and oscillatory behavior in Euclidean settings.
Extends harmonic analysis to topological groups and abstract algebraic structures.
Studies transforms such as Laplace and Fourier and operator techniques for solving equations.
Investigates equations where unknown functions appear under integral operators.
Studies linear and nonlinear operators on function spaces and their spectral properties.
Develops methods for extremizing functionals and controlling dynamic systems under constraints.
This area explores spaces, shapes, continuity, and the geometric structure of mathematical objects.
Studies approximation methods, series, and expansions of functions.
Studies vector spaces of functions and operators acting on them.
Studies convex bodies, polytopes, and geometric combinatorics in discrete and continuous settings.
Examines geometric structures, transformations, and spatial relationships.
Studies curves, surfaces, and manifolds through calculus and geometric structure.
Studies continuity, compactness, connectedness, and abstract topological spaces.
Uses algebraic methods to study topological spaces and their invariants.
Studies higher-dimensional spaces built from local Euclidean pieces.
Combines analysis and geometry on manifolds, including differential operators and global invariants.
This area develops models for randomness, inference, optimization, and the mathematical treatment of data and uncertainty.
Formalizes randomness and the evolution of uncertain systems over time.
Develops methods for collecting, summarizing, and drawing inference from data.
Studies algorithms and approximations for solving mathematical problems computationally.
Studies the mathematical foundations of computation, algorithms, and information processing.
Models motion and interactions of mechanical systems using differential equations and variational principles.
Studies elasticity, plasticity, and continuum models for solid materials under stress.
Analyzes fluids in motion and at rest, including flow stability and conservation laws.
Applies mathematical models to wave propagation, optics, and electromagnetic fields.
Studies energy, entropy, diffusion, and mathematical models of thermal processes.
Provides mathematical frameworks for quantum states, observables, and dynamics.
Connects microscopic interactions with macroscopic behavior through probabilistic models.
Studies spacetime geometry and gravitational phenomena using geometric and analytic methods.
Applies mathematics to celestial dynamics, radiation processes, and large-scale cosmic systems.
Models Earth systems such as seismic waves, fluid interiors, and geodynamic processes.
Applies mathematics to optimization, planning, and decision-making.
Uses mathematical models to analyze strategic interaction and decision-making.
Applies mathematics to biological systems and scientific modeling.
Studies the behavior and control of dynamic systems.
Applies mathematical methods to communication systems and circuit design.
Studies teaching, learning, curriculum, and assessment in mathematics across educational settings.
This continuation highlights five representative subtopics that sit beneath the broader topic classes already listed.
Collects foundational notes on the philosophy, methodology, and broad applications of mathematics.
Surveys historical developments, biographies, and the cultural context of key mathematical ideas.
Focuses on formal systems, proof structures, and the basic machinery of logical reasoning.
Deals with counting, arranging, and classifying finite objects in structured ways.
Explores partial orders, total orders, and hierarchies that underpin many mathematical structures.