Lists Of Mathematics Topics

Lists of Mathematics Topics

The vast and often intimidating landscape of mathematics is a sprawling testament to human intellect, curiosity, and, let's be honest, a penchant for making things far more complex than they initially appear. These comprehensive collections, conveniently compiled into various lists of mathematics topics, serve as a rather necessary navigational tool through this intellectual labyrinth. While some of these lists meticulously link to hundreds of intricate articles, others, with a charming brevity, connect to only a select few. The template that follows this introductory overview provides direct access to alphabetical compilations of virtually all mathematical articles available. However, the true utility of this particular article lies in its deliberate reorganization of this same content, presenting it in a structured manner that is, presumably, better suited for browsing by those who prefer a semblance of order in their universe.

These lists, with their often bewildering diversity, endeavor to encompass virtually every conceivable facet of both foundational and highly advanced mathematics. They delve into the very methodology of mathematical thought, articulate a plethora of profound mathematical statements, catalog an exhaustive array of integrals, elucidate overarching general concepts, categorize the myriad forms of mathematical objects, and provide invaluable reference tables. Beyond the purely conceptual and abstract, these lists also meticulously document equations named after people, detail the various societies that foster mathematical endeavor, honor the countless mathematicians who have shaped the field, provide access to a wealth of journals, and even offer meta-lists for those who enjoy lists about lists.

It is crucial to understand that the fundamental purpose of this compilation of lists diverges significantly from the meticulously structured Mathematics Subject Classification (MSC) system. The MSC, a formidable hierarchical scheme, was meticulously formulated and is rigorously maintained by the venerable American Mathematical Society. Indeed, it is a standard practice that many mathematics journals, when soliciting research papers and expository articles, require authors to precisely list subject codes drawn from the MSC. These subject codes, once assigned and published, are then diligently utilized by the two preeminent reviewing databases in the mathematical world: Mathematical Reviews and Zentralblatt MATH. This present list, however, embraces a more eclectic approach, featuring numerous items that simply would not, and frankly could not, comfortably fit within such a rigid classification framework. Consider, for instance, the inclusion of a list of exponential topics or a list of factorial and binomial topics—their sheer breadth and the unexpected diversity of their coverage often manage to surprise even the most jaded reader.

Basic Mathematics

This branch, often considered the entry point into the more profound depths of mathematical thought, is typically encountered during secondary education or, for the more ambitious, within the rigorous curriculum of the first year of university. It lays the groundwork, the very bedrock upon which all subsequent mathematical edifices are constructed, and without which, one would simply be lost.

Outline of arithmetic: The elemental art of counting and calculating, where numbers first learn to interact. A foundational skill, if you insist on calling it that.
Outline of discrete mathematics: The study of distinct, separate values, a refreshing change from the endless continuity. Crucial for understanding computation, or so they say.
List of calculus topics: Where numbers finally learn to move and change, dealing with rates and accumulations. The gateway drug to analysis.
List of geometry topics: The study of shapes, sizes, positions, and properties of space. Where mathematics first became visual, before it became abstract and difficult.
Outline of geometry: A structured overview of the aforementioned spatial concerns.
List of trigonometry topics: The intricate dance of angles and sides in triangles, particularly useful for making sense of waves and other periodic annoyances.
Outline of trigonometry: A concise framework for mastering the trigonometric arts.
List of trigonometric identities: The fundamental equalities that make trigonometric calculations less of a headache, or more, depending on your disposition.
List of logarithmic identities: The rules governing logarithms, tools for simplifying multiplication into addition, a testament to humanity's eternal quest for easier ways to do things.
List of integrals of logarithmic functions: Specific solutions for integrating functions involving logarithms, for when you absolutely must find the area under that particular curve.
List of set identities and relations: The foundational rules governing collections of objects, the very building blocks of modern mathematics. Because everything is a set, eventually.
List of topics in logic: The study of valid reasoning and inference, a pursuit that often seems to elude actual human discourse.

Areas of Advanced Mathematics

As a perhaps overly simplistic guide, this extensive list is broadly bifurcated into sections dedicated to pure mathematics and applied mathematics. One might imagine these as two distinct kingdoms, yet in the messy reality of intellectual pursuit, these branches are far from isolated; they are, in truth, deeply overlapping, perpetually intertwined, and constantly borrowing concepts, techniques, and even existential crises from one another.

Pure mathematics

Main article: Pure mathematics

This is where mathematics, unburdened by the immediate demands of the physical world, often delves into its deepest, most abstract, and some might say, most beautiful forms. Others might say it's where mathematicians go to escape reality.

Algebra

Algebra, in its essence, transcends the mere manipulation of symbols and variables you might recall from secondary school. It is, in fact, the profound and systematic study of what are known as algebraic structures. These structures are typically composed of sets—collections of objects, if you must—upon which specific operations are meticulously defined, and which, in turn, must satisfy a precise set of foundational axioms. The vast and intricate field of algebra is subsequently subdivided with remarkable precision, primarily according to the specific type of structure under scrutiny. For example, group theory, a cornerstone of modern algebra, is exclusively concerned with a particular, elegantly defined algebraic structure known as a group. This rigorous focus allows for deep insights into symmetry, transformations, and the very fabric of mathematical order.

Outline of algebra: A structural overview of this fundamental branch.
Glossary of field theory: The specialized vocabulary for fields, which are sets where addition, subtraction, multiplication, and division behave exactly as you'd expect.
Glossary of group theory: The language of groups, those elegant structures of symmetry and transformation.
Glossary of linear algebra: The precise terminology for vectors, matrices, and the transformations that underpin much of modern science and computation.
Glossary of ring theory: The lexicon of rings, structures where addition and multiplication are well-behaved, but division might be a bit rude.
List of abstract algebra topics: A dive into the more generalized and formal aspects of algebraic structures.
List of algebraic structures: A catalogue of the diverse frameworks that define algebra.
List of Boolean algebra topics: The algebra of truth values, the foundation of digital logic and, arguably, common sense.
List of category theory topics: A highly abstract realm, studying relationships between mathematical structures themselves. Because why study objects when you can study the relationships between the objects?
List of cohomology theories: Advanced tools for probing the 'holes' and topological properties of spaces, often bridging algebra and topology.
List of commutative algebra topics: Focusing on rings where multiplication doesn't care about order. How polite.
List of homological algebra topics: Bridging algebra and topology, using chains and cycles to understand structures.
List of group theory topics: Specific areas within the study of groups.
List of representation theory topics: How abstract algebraic structures can be 'represented' by more concrete linear transformations.
List of linear algebra topics: A comprehensive collection of concepts related to vectors, matrices, and linear mappings.
List of reciprocity laws: Elegant statements about the relationships between different number-theoretic properties.

Calculus and analysis

Fourier series approximation of square wave in five steps.

Calculus, a veritable cornerstone of modern mathematics, is fundamentally dedicated to the precise computation of limits, derivatives, and integrals of functions, particularly those involving real numbers. It is the language through which we grapple with the concepts of instantaneous rates of change, the accumulation of quantities, and the behavior of dynamic systems. From this foundational bedrock of calculus, the more abstract and expansive field of analysis meticulously evolved. Analysis extends these concepts far beyond the familiar confines of real numbers, delving into the properties of functions, sequences, and series in more abstract spaces, often employing rigorous epsilon-delta arguments to establish its truths. It seeks to understand continuity, convergence, and the very fabric of mathematical functions with an almost obsessive precision.

Glossary of tensor theory: The specialized vocabulary for tensors, which are generalized vectors, crucial for describing physical phenomena in relativity and fluid dynamics.
List of complex analysis topics: Extending calculus and analysis to the realm of complex numbers, revealing unexpected symmetries and profound connections.
List of functional analysis topics: The study of vector spaces of functions, often infinite-dimensional, a crucial tool in quantum mechanics and differential equations.
List of vector spaces in mathematics: A catalogue of structures that generalize the concept of Euclidean space.
List of integration and measure theory topics: The rigorous foundation for integration, extending beyond the Riemann integral to more general sets and functions.
List of harmonic analysis topics: Decomposing functions into simpler components, like waves, to understand their underlying structure.
List of Fourier analysis topics: Specific applications and theories arising from the decomposition of functions into sinusoidal waves, indispensable in signal processing.
List of mathematical series: Infinite sums, their convergence, and their often astonishing results.
List of multivariable calculus topics: Extending the concepts of calculus to functions of several variables, essential for understanding higher dimensions.
List of q-analogs: Generalizations of mathematical expressions involving a parameter 'q', often linking to quantum theory and combinatorics.
List of real analysis topics: The rigorous study of real numbers, sequences, limits, continuity, and differentiation, the true bedrock of calculus.
List of variational topics: Dealing with problems of optimizing functionals, finding paths that minimize or maximize certain quantities.
See also Dynamical systems and differential equations section below for related concepts.

Geometry and topology

Ford circles—A circle rests upon each fraction in lowest terms. Each touches its neighbors without crossing.

Geometry, in its most ancient and intuitive form, began as the meticulous study of spatial figures—the familiar circles, squares, cubes, and spheres that populate our immediate physical world. However, over millennia, this field has undergone a profound and considerable generalization, expanding far beyond these initial concrete forms. From this rich geometric lineage, topology emerged as a distinct, yet intimately related, discipline. Topology takes a more abstract view, focusing on those intrinsic properties of geometric figures and spaces that remain invariant, that do not fundamentally change, even when the figures are subjected to continuous deformations such as stretching, bending, or twisting. It's the study of shape without rigid form, where a coffee cup is topologically equivalent to a donut, both possessing a single 'hole'. This shift in perspective allows for a deeper understanding of connectivity, compactness, and dimension, moving beyond mere measurement to the fundamental nature of space itself.

Glossary of differential geometry and topology: The specialized language for studying smooth shapes and spaces using calculus.
Glossary of general topology: The lexicon for the most abstract study of space, where 'nearness' is defined without a metric.
Glossary of Riemannian and metric geometry: The terminology for geometries where distances and curvatures are central, often linked to general relativity.
Glossary of scheme theory: A highly abstract and foundational language in algebraic geometry, extending the concept of algebraic varieties.
List of algebraic geometry topics: The fascinating intersection of algebra and geometry, using polynomial equations to describe geometric objects.
List of algebraic surfaces: Specific examples of surfaces defined by algebraic equations.
List of algebraic topology topics: Using algebraic tools to classify and understand topological spaces.
List of cohomology theories: (Repeated, indicating its cross-disciplinary nature) Advanced algebraic tools for probing the 'holes' and topological properties of spaces.
List of circle topics: Everything you ever wanted to know about circles, and probably more.
List of topics related to pi: The ubiquitous constant that appears in everything from circles to quantum mechanics.
List of curves topics: A comprehensive collection of different types of curves and their properties.
List of differential geometry topics: The study of smooth curves, surfaces, and manifolds using the tools of calculus.
List of general topology topics: The fundamental concepts of open sets, closed sets, continuity, and compactness.
List of geometric shapes: A visual catalogue of the forms that populate our world and mathematical thought.
List of geometric topology topics: The study of low-dimensional manifolds and their embeddings, often more visual than algebraic topology.
List of geometry topics: A general overview of the field.
List of knot theory topics: The mathematical study of knots, not just for Boy Scouts, but for understanding DNA and fundamental particles.
List of Lie group topics: Continuous groups of symmetries, crucial in physics and differential geometry.
List of mathematical properties of points: The characteristics and classifications of points within various mathematical contexts.
List of topology topics: A general overview of the field.
List of topologies: Different ways to define 'open sets' on a given set, leading to different notions of continuity and convergence.
Topological property: Characteristics that remain invariant under continuous deformations.
List of triangle topics: The humble triangle, a deceptively simple shape with an astonishing depth of properties and theorems.

Combinatorics

Combinatorics is a vibrant and often deceptively complex branch of mathematics primarily concerned with the study of discrete objects—entities that are distinct and separable, rather than continuous—and these objects are, more often than not, finite in number. This expansive field grapples with a variety of fundamental aspects. It includes "counting" the objects that satisfy specific, often intricate, criteria, a pursuit known as enumerative combinatorics. It also involves the profound challenge of deciding precisely when such criteria can actually be met, and, when they can, the art of constructing and meticulously analyzing the objects that indeed fulfill these criteria, as exemplified in the elegant theories of combinatorial designs and the abstract framework of matroid theory. Furthermore, combinatorics is dedicated to the quest for "largest," "smallest," or "optimal" objects, a domain explored by extremal combinatorics and combinatorial optimization. Finally, it investigates the potential algebraic structures that these diverse objects might possess, leading to the rich subfield of algebraic combinatorics. In essence, combinatorics is about understanding structure, arrangement, and possibility in a world of distinct elements.

Outline of combinatorics: A structured overview of this intricate field.
Glossary of graph theory: The specialized language for graphs, which are mathematical structures used to model pairwise relations between objects.
List of graph theory topics: A comprehensive collection of concepts related to graphs, from paths and cycles to coloring and networks.

Logic

Venn diagrams are illustrations of set theoretical, mathematical or logical relationships.

Logic, in its most fundamental sense, serves as the indispensable foundation that underpins not only mathematical logic but, by extension, the entire edifice of mathematics itself. Its primary endeavor is to rigorously formalize what constitutes valid reasoning, transforming intuitive notions of truth and inference into precise, axiomatic systems. More specifically, it painstakingly attempts to define, with unwavering clarity, the exact criteria that constitute a legitimate mathematical proof. This quest for formal rigor ensures that mathematical statements, once proven, stand as immutable truths, free from ambiguity or subjective interpretation. It's the meticulous plumbing of the mathematical house, ensuring everything flows correctly.

List of Boolean algebra topics: (Repeated, again highlighting its cross-disciplinary importance) The algebra of truth values, essential for formal logic and computation.
List of first-order theories: Formal systems of logic that allow quantification over individuals but not over properties or relations.
List of large cardinal properties: Concepts in set theory dealing with extremely large infinite numbers, pushing the boundaries of what can be consistently defined.
List of mathematical logic topics: A comprehensive overview of the intersection of logic and mathematics.
List of set theory topics: Concepts related to the fundamental theory of collections of objects.
Glossary of order theory: The specialized vocabulary for studying mathematical relations that capture the intuitive notion of order or comparison.

Number theory

The branch of mathematics deals with the properties and relationships of numbers, especially positive integers. Number theory stands as a venerable branch of pure mathematics, dedicated with almost singular focus to the profound study of integers—the whole numbers, both positive and negative, along with zero—and the intricate relationships governing integer-valued functions. The esteemed German mathematician Carl Friedrich Gauss, a figure whose intellectual shadow stretches long over the landscape of mathematics, famously declared, with an air of undeniable authority, that "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." This declaration underscores the field's intrinsic beauty, its foundational importance, and its ability to pose questions of astonishing depth from seemingly simple premises. Number theory also, quite naturally, extends its gaze to the natural numbers, which are the positive integers (or sometimes non-negative integers, depending on one's convention). One of the most central and enduring concepts within number theory is that of the prime number—a natural number greater than 1 that has no positive divisors other than 1 and itself. The study of primes has captivated mathematicians for millennia, giving rise to countless questions that, despite their apparent simplicity, have consistently resisted resolution, continuing to elude even the most brilliant minds and serving as a wellspring of ongoing mathematical research.

List of algebraic number theory topics: Extending number theory to algebraic numbers, which are roots of polynomial equations.
List of number theory topics: A broad collection of concepts within the study of integers and their properties.
List of recreational number theory topics: The playful, yet often profound, side of number theory, exploring puzzles and patterns.
Glossary of arithmetic and Diophantine geometry: The specialized language for studying integer solutions to polynomial equations using geometric methods.
List of prime numbers—not just a mundane table, but a fascinating compilation of various kinds of prime numbers (each thoughtfully accompanied by an illustrative table), revealing their diverse classifications and elusive patterns.
List of zeta functions: Complex functions with deep connections to prime numbers and analytic number theory, most famously the Riemann zeta function.

Applied mathematics

This is where mathematics, with a visible sigh of reluctance, condescends to engage with the messy, imperfect realities of the physical world. It's where abstract tools are forged into instruments for understanding, predicting, and, occasionally, even controlling the universe around us.

Dynamical systems and differential equations

Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator.

A differential equation is, at its core, an equation that involves an unknown function alongside one or more of its derivatives. These mathematical constructs are indispensable for describing phenomena where rates of change are paramount, forming the bedrock for modeling everything from the trajectory of a projectile to the spread of a disease.

In the realm of a dynamical system, a meticulously defined, fixed rule dictates the precise time dependence of a point within a specified geometrical space. Such systems provide a powerful framework for understanding how states evolve over time. Consider, for instance, the elegant mathematical models painstakingly developed to describe the rhythmic swinging of a clock pendulum, the complex, turbulent flow of water through a pipe, or the fluctuating population dynamics of fish each spring within a serene lake. These are all quintessential examples of dynamical systems, demonstrating how seemingly simple rules can give rise to extraordinarily rich and often chaotic behaviors. They offer insights into stability, unpredictability, and the very arrow of time within mathematical frameworks.

List of dynamical systems and differential equations topics: A comprehensive collection covering the theory and applications of systems that evolve over time.
List of nonlinear partial differential equations: Equations where the principle of superposition does not hold, leading to complex and often unpredictable behaviors.
List of partial differential equation topics: Equations involving multiple independent variables and their partial derivatives, crucial for modeling wave propagation, heat flow, and electromagnetism.

Mathematical physics

Mathematical physics is a specialized and intellectually rigorous discipline concerned primarily with "the astute application of mathematics to address the myriad problems encountered in physics, and simultaneously, the dedicated development of sophisticated mathematical methods that are not only suitable for such intricate applications but also essential for the precise and elegant formulation of physical theories." This symbiotic relationship means that advances in one field often spur profound breakthroughs in the other, pushing the boundaries of both our mathematical understanding and our comprehension of the cosmos. It's where the abstract beauty of mathematics is forced to confront the sometimes inconvenient truths of reality.

List of mathematical topics in classical mechanics: The mathematical formalisms behind Newton's laws, Lagrangian and Hamiltonian mechanics, describing the motion of macroscopic objects.
List of mathematical topics in quantum theory: The abstract mathematical structures—Hilbert spaces, operators, group theory—that underpin the bizarre world of subatomic particles.
List of mathematical topics in relativity: The differential geometry and tensor calculus required to describe spacetime curvature and gravitational phenomena.
List of string theory topics: The highly theoretical and mathematically intensive concepts underpinning the idea that fundamental particles are tiny vibrating strings.
Index of wave articles: A collection of topics related to waves, a ubiquitous phenomenon described by differential equations across various branches of physics.

Theory of computation

Ray tracing is a process based on computational mathematics.

The intertwined fields of mathematics and computing intersect in a myriad of profound and practical ways. This confluence is evident both in computer science, which is fundamentally the systematic study of algorithms and data structures, and in scientific computing, an equally vital discipline dedicated to the development and application of algorithmic methods for the efficient and accurate resolution of complex problems encountered across mathematics, science, and engineering. The theory of computation provides the rigorous mathematical framework necessary to understand the limits and capabilities of what can be computed, defining the very boundaries of the digital realm.

List of algorithm general topics: A broad overview of algorithms, the step-by-step procedures for solving computational problems.
List of computability and complexity topics: Exploring what problems can be solved by algorithms and how efficiently they can be solved.
Lists for computational topics in geometry and graphics:
- List of combinatorial computational geometry topics: Dealing with the discrete structures and arrangements of geometric objects in computational settings.
- List of computer graphics and descriptive geometry topics: The mathematical foundations for rendering images and representing 3D objects.
- List of numerical computational geometry topics: Using numerical methods to solve geometric problems, often involving approximations.
List of computer vision topics: The mathematical and algorithmic methods for enabling computers to 'see' and interpret images.
List of formal language and literal string topics: The theoretical underpinnings of programming languages and symbolic manipulation.
List of numerical analysis topics: The study of algorithms for obtaining approximate numerical solutions to mathematical problems, often essential for problems without exact solutions.
List of terms relating to algorithms and data structures: A lexicon for the fundamental building blocks of computation.

Information theory and signal processing

Information theory is a sophisticated branch of applied mathematics and, intriguingly, also a significant component of social science, primarily concerned with the rigorous quantification of information. Historically, this field was meticulously developed with the crucial objective of identifying the fundamental limits inherent in compressing and reliably communicating data across noisy channels. It provides the mathematical framework for understanding the very essence of data, its entropy, and its capacity to be transmitted.

Signal processing, an equally vital discipline, focuses on the systematic analysis, interpretation, and manipulation of various signals. The signals of interest span an incredibly broad spectrum, encompassing everything from audible sound waves and intricate images to subtle biological signals such as an ECG (electrocardiogram), the echoes of radar signals, and a multitude of other forms. The processing of such diverse signals involves a suite of advanced techniques, including sophisticated filtering to remove unwanted components, efficient storage and precise reconstruction methodologies, the challenging task of separating meaningful information from pervasive noise, rigorous data compression to reduce data volume, and the critical process of feature extraction to identify salient characteristics.

List of algebraic coding theory topics: Using algebraic structures to design error-correcting codes, ensuring reliable data transmission.
List of information theory topics: A comprehensive overview of entropy, channel capacity, and coding theory.
List of cryptography topics: The mathematical principles behind secure communication, ensuring privacy and authenticity.

Probability and statistics

The "bell curve"—the probability density function of the normal distribution.

Main article: Lists of statistics topics

Probability theory is the meticulous formalization and systematic study of the mathematics governing uncertain events or knowledge. It provides a rigorous framework for quantifying chance, risk, and the likelihood of various outcomes, transforming intuitive notions of randomness into precise mathematical models. The closely related field of mathematical statistics then takes these probabilistic foundations and develops the overarching statistical theory with an equally rigorous mathematical approach. It is, in essence, the theoretical backbone that underpins all statistical inference.

It is important to note, however, that Statistics itself, as an autonomous discipline, is the science primarily concerned with the methodical collection, insightful analysis, rigorous interpretation, presentation, and organization of data. While it heavily relies on applied mathematics and probability, it stands as a distinct field with its own methodologies and objectives, often tackling real-world problems directly where mathematical statistics provides the theoretical tools.

Catalog of articles in probability theory: A comprehensive listing of topics within the realm of chance.
List of probability topics: Specific concepts and theorems in probability.
List of stochastic processes topics: The study of random phenomena evolving over time, from stock prices to particle movements.
List of probability distributions: The mathematical functions that describe the likelihood of different outcomes for a random variable.
List of statistics topics: A broad collection of concepts in data analysis and inference.
Outline of regression analysis: A structured overview of techniques for modeling relationships between variables.

Game theory

Game theory is a fascinating branch of mathematics that employs sophisticated models to meticulously study strategic interactions among rational decision-makers, typically within formalized incentive structures, affectionately (or perhaps ironically) termed "games." Far from being confined to mere parlor games, its profound insights have found widespread and crucial applications across an astonishingly diverse array of fields. These include the intricate world of economics, the study of human societies and cultures in anthropology, the dynamics of power and governance in political science, the complexities of human interaction in social psychology, and even the high-stakes realm of military strategy. It's the mathematics of conflict and cooperation, where everyone tries to outsmart everyone else, usually with predictable results.

Glossary of game theory: The specialized vocabulary for understanding strategic interactions.
List of games in game theory: A catalogue of classic scenarios, from the Prisoner's Dilemma to Chicken, illustrating various strategic interactions.

Operations research

Operations research is a highly interdisciplinary field dedicated to the systematic study and practical application of sophisticated mathematical models, advanced statistical methods, and powerful algorithms. Its overarching goal is to provide robust aid in complex decision-making processes, typically with the ultimate objective of significantly improving or rigorously optimizing the performance of intricate real-world systems. Whether it's streamlining supply chains, scheduling complex projects, or allocating scarce resources, operations research seeks the most efficient path through a labyrinth of constraints, often finding solutions that defy intuitive human judgment.

List of knapsack problems: A class of optimization problems where one must choose the most valuable items to fit into a limited capacity.
List of network theory topics: The study of graphs that represent connections between entities, crucial for understanding communication, transport, and social networks.

Methodology

These lists delve into the underlying approaches and systematic procedures employed across various mathematical disciplines, showcasing the diverse tools and frameworks that mathematicians use to construct knowledge and solve problems.

List of graphical methods: Techniques that use visual representations to analyze data or solve mathematical problems.
List of mathematics-based methods: General approaches and techniques rooted in mathematical principles, applicable across various domains.
List of rules of inference: The logical forms that guarantee the truth of a conclusion given the truth of its premises, fundamental to constructing proofs.

Mathematical statements

A mathematical statement is, at its core, a precise proposition or an unequivocal assertion of some mathematical fact, a formal declaration, or a meticulously constructed mathematical entity. Such statements encompass a broad spectrum of declarations, ranging from the fundamental axioms—the self-evident truths accepted without proof—to the profound theorems that may be rigorously proved from these axioms. They also include conjectures, which are intriguing propositions that remain unproven, or, in some cases, even unprovable, yet continue to inspire intense mathematical investigation. Furthermore, this category includes algorithms, which are step-by-step procedures designed for computing the answers to questions that can be articulated with mathematical precision. These statements form the very language and content of mathematical discourse.

List of algorithms: Step-by-step procedures for computation, the backbone of all digital operations.
List of axioms: Fundamental truths accepted without proof, the starting points of mathematical systems.
List of conjectures: Unproven mathematical statements, the challenges that drive research forward.
List of conjectures by Paul Erdős: A collection of the famously prolific mathematician's open problems.
Combinatorial principles: Fundamental rules for counting and arrangement, underlying much of combinatorics.
List of equations: Mathematical expressions asserting the equality of two quantities, from simple identities to complex differential equations.
List of formulae involving pi: The myriad ways the constant π appears in mathematical expressions.
List of representations of e: The various mathematical forms and series that define Euler's number.
List of inequalities: Mathematical statements comparing the relative size or order of two quantities.
List of lemmas: Minor theorems, often proved as stepping stones to more significant results.
List of mathematical identities: Equations that hold true for all values of their variables.
List of mathematical proofs: Examples of rigorous demonstrations of mathematical truths.
List of theorems: Proven mathematical statements, representing established knowledge.

General concepts

This section gathers a selection of foundational ideas and overarching themes that permeate various branches of mathematics. These are the conceptual threads that weave through different disciplines, offering common language and abstract frameworks for understanding diverse phenomena.

List of convexity topics: The study of convex sets and functions, crucial in optimization and geometry.
List of dualities: Concepts where two distinct mathematical objects or theories are related in a way that allows for interchangeability.
List of exceptional set concepts: Ideas related to sets that are "small" or "negligible" in some mathematical sense.
List of exponential topics: Concepts related to exponential functions and growth.
List of factorial and binomial topics: The mathematics of permutations, combinations, and counting arrangements.
List of fractal topics: The study of infinitely self-similar patterns, exhibiting complexity at every scale.
List of logarithm topics: Concepts related to logarithms, the inverse of exponential functions.
List of mathematical properties of points: (Repeated, indicating its fundamental nature) Characteristics of points within various mathematical spaces.
List of numeral system topics: Different ways of representing numbers.
List of order topics: Concepts related to ordering and comparisons within sets.
List of partition topics: Ways to break down a number or set into smaller, non-overlapping parts.
List of permutation topics: The study of arrangements of objects where order matters.
List of polynomial topics: Concepts related to polynomials, fundamental algebraic expressions.
List of properties of sets of reals: Characteristics unique to subsets of real numbers, such as compactness or connectedness.
List of transforms: Mathematical operations that convert functions from one domain to another, often simplifying analysis.

Mathematical objects

Among the countless entities that populate the abstract universe of mathematics are a diverse array of "objects." These are not tangible items, of course, but rather rigorously defined conceptual constructs that serve as the subjects of mathematical study and manipulation. They include the fundamental numbers that underpin all quantitative thought, the dynamic functions that describe relationships and transformations, the foundational sets that group elements, and an almost bewildering variety of entities referred to as "spaces" of one kind or another—each endowed with specific properties. This category also encompasses intricate algebraic structures such as rings, groups, or fields, each with its own set of axioms and behaviors. And, of course, there are countless other, often highly specialized, things that mathematicians invent and explore, pushing the boundaries of what can be formally defined and understood.

List of mathematical examples: Concrete instances used to illustrate theories or counter-examples.
List of algebraic surfaces: (Repeated, showing its cross-listing) Surfaces defined by polynomial equations.
List of curves: Various one-dimensional geometric objects.
List of complex reflection groups: Groups generated by reflections in complex vector spaces.
List of complexity classes: Categorizations of computational problems by the resources (time, memory) required to solve them.
List of examples in general topology: Illustrative cases that clarify abstract topological concepts.
List of finite simple groups: The fundamental building blocks of all finite groups, a monumental classification project.
List of Fourier-related transforms: Variations on the Fourier transform, used for signal analysis.
List of manifolds: Spaces that locally resemble Euclidean space, crucial in geometry and physics.
List of mathematical constants: Numbers like π and e that appear repeatedly in mathematical formulas.
List of mathematical functions: Rules that assign outputs to inputs, the workhorses of calculus and analysis.
List of mathematical knots and links: Specific examples of mathematical knots and their generalizations.
List of mathematical shapes: A catalogue of geometric forms.
List of mathematical spaces: Various abstract settings where mathematical objects reside and interact.
List of matrices: Rectangular arrays of numbers, fundamental in linear algebra and transformations.
List of numbers: The basic elements of arithmetic, from natural numbers to complex numbers and beyond.
List of polygons, polyhedra and polytopes: Regular and irregular geometric figures in various dimensions.
List of regular polytopes: Highly symmetric geometric objects, generalizations of regular polygons and polyhedra.
List of simple Lie groups: The fundamental building blocks of all Lie groups, crucial for understanding continuous symmetries.
List of small groups: A classification of groups with a small number of elements.
List of special functions and eponyms: Functions that arise frequently in physics and engineering, often named after their discoverers.
List of surfaces: Two-dimensional manifolds embedded in higher dimensions.
Table of Lie groups: A structured summary of these important continuous symmetry groups.

Equations named after people

It seems humanity has an insatiable need to immortalize itself, and mathematicians are no exception. This list serves as a testament to that particular vanity.

Scientific equations named after people: A broad collection of fundamental equations across science and mathematics, each bearing the name of its discoverer or prominent developer.

About mathematics

Beyond the abstract symbols and rigorous proofs, mathematics also has a rich history, a vibrant community, and a significant cultural impact. These lists delve into the human and societal aspects of the discipline.

List of letters used in mathematics and science: The alphabet soup that makes up mathematical notation.
List of mathematical societies: Organizations dedicated to the advancement and dissemination of mathematical knowledge.
List of mathematics competitions: Challenges designed to test and foster mathematical talent.
List of mathematics history topics: The chronological development and evolution of mathematical ideas.
List of publications in mathematics: Significant books, monographs, and other published works.
List of mathematics journals: Periodicals for the dissemination of new research and scholarly articles.

Mathematicians

Main article: List of mathematicians

Mathematicians—those peculiar individuals who choose to dedicate their lives to the study and rigorous research across all the diverse and often bewildering areas of mathematics. The relentless pace of discovery and the publication of novel insights in mathematics continues at an immense and frankly exhausting rate, filling hundreds of specialized scientific journals. Many of these periodicals are, naturally, exclusively devoted to pure mathematics, while a substantial number are dedicated to the myriad subjects to which mathematics is applied, such as theoretical computer science and theoretical physics. It's a never-ending intellectual marathon, apparently.

List of films about mathematicians: Cinematic portrayals of these often eccentric, sometimes brilliant, individuals.
List of game theorists: Those who analyze strategic interactions and decision-making.
List of geometers: Experts in the study of space and shape.
List of logicians: Those who formalize reasoning and the foundations of mathematics.
List of mathematicians: A comprehensive catalogue of individuals who have contributed to the field.
List of mathematical probabilists: Specialists in the mathematics of chance and uncertainty.
List of statisticians: Experts in the collection, analysis, interpretation, and presentation of data.

Work of particular mathematicians

Even mathematicians, it seems, crave a legacy beyond their lifespan. This section is a testament to the enduring impact of individual minds, cataloging the theorems, concepts, and objects that bear their names.

Reference tables

For those who prefer their information pre-digested and neatly organized. A necessary evil for quick lookups, I suppose.

List of mathematical reference tables: Compilations of common values, formulas, and properties.
List of moments of inertia: Tabulated values for the rotational inertia of various shapes, crucial in physics and engineering.
Table of derivatives: A quick reference for the derivatives of common functions.

Integrals

In calculus, the integral of a function is a powerful and versatile concept that serves as a generalization for fundamental ideas such as area under a curve, total mass, volume of a solid, the sum of infinitely many infinitesimally small quantities, and a wide array of other cumulative totals. It is the inverse operation to differentiation, allowing us to reconstruct a function from its rate of change. The following pages meticulously list the integrals of a vast array of different functions, providing essential tools for solving problems in physics, engineering, economics, and virtually any field where accumulation or total change is a concern.

Lists of integrals: A meta-list for all integral types.
List of integrals of exponential functions: Specific solutions for integrating functions involving the exponential e.
List of integrals of hyperbolic functions: Integrals for functions analogous to trigonometric functions but defined on a hyperbola.
List of integrals of inverse hyperbolic functions: Solutions for the integrals of the inverse forms of hyperbolic functions.
List of integrals of inverse trigonometric functions: Integrals for the arcsin, arccos, etc. functions.
List of integrals of irrational functions: Solutions for functions involving roots and non-integer powers.
List of integrals of logarithmic functions: (Repeated, indicating its importance) Specific solutions for integrating functions involving logarithms.
List of integrals of rational functions: Solutions for integrating functions that are ratios of polynomials.
List of integrals of trigonometric functions: Solutions for integrating sine, cosine, tangent, and their variants.

Journals

The lifeblood of ongoing mathematical research, where new discoveries are announced and debated, often in prose only comprehensible to a select few.

List of mathematics journals: Periodicals for the dissemination of new research and scholarly articles in mathematics.
List of mathematics education journals: Publications focused on the pedagogy and learning of mathematics.
Category:History of science journals: Journals that document the historical development of scientific and mathematical thought.
Category:Philosophy of science literature: Publications exploring the philosophical underpinnings and implications of scientific inquiry.

Meta-lists

Because sometimes, a list isn't enough. You need a list about lists, or a glossary to understand the glossaries. It's an ouroboros of organization.

Glossary of mathematical symbols: A dictionary for the cryptic hieroglyphs mathematicians use.
List of important publications in mathematics: Canonical works that have shaped the field.
List of important publications in statistics: Key texts that have defined statistical theory and practice.
List of mathematical theories: The grand frameworks and conceptual systems within mathematics.
List of mathematics categories: A classification of mathematical topics into broader areas.
List of mathematical symbols by subject: Symbols organized by the mathematical area they typically appear in.
Table of logic symbols: A quick reference for the symbols used in formal logic.
Table of mathematical symbols: A general reference for commonly used mathematical notation.

Others

More lists, because why not.

Notes

^Note 1 : This definition is directly from the Journal of Mathematical Physics [1].

External links and references

The 2000 Mathematics Subject Classification from the American Mathematical Society, a rigorous scheme that authors are often asked to use to classify their submissions to mathematics research journals; those articles that are subsequently published then include these classifications for indexing purposes.
The Mathematical Atlas
Maths Formula
PlanetMath
Paul's Online Notes
ProofWiki
The Encyclopedia of Mathematics