Notes on mathematics organized loosely by the Mathematics Subject Classification (MSC 2020). Each number is a separate book outlining that area.
Books in progress
| # | Title | About |
|---|---|---|
| 00 | General Mathematics | Language, structure, methodology, and cross-cutting tools across all branches |
| 01 | History and Biography | How mathematical ideas emerged, stabilized, and influenced later work |
| 03 | Logic and Foundations | Formal logic, set theory, computability, proof theory |
| 05 | Combinatorics | Counting, arrangement, structure, and extremal behavior of finite systems |
| 06 | Order and Lattices | Partially ordered sets, lattices, and ordered algebraic structures |
| 08 | General Algebraic Systems | Operations and identities as a unifying framework for all algebraic theories |
| 11 | Number Theory | Integers, primes, modular arithmetic, Diophantine equations |
Full MSC index
Numbers without a link are planned but not started yet.
| # | Area |
|---|---|
| 00 | General and overarching topics |
| 01 | History and biography |
| 03 | Mathematical logic and foundations |
| 05 | Combinatorics |
| 06 | Order, lattices, ordered algebraic structures |
| 08 | General algebraic systems |
| 11 | Number theory |
| 12 | Field theory and polynomials |
| 13 | Commutative algebra |
| 14 | Algebraic geometry |
| 15 | Linear and multilinear algebra; matrix theory |
| 16 | Associative rings and algebras |
| 17 | Non-associative rings and algebras |
| 18 | Category theory; homological algebra |
| 19 | K-theory |
| 20 | Group theory and generalizations |
| 22 | Topological groups, Lie groups |
| 26 | Real functions |
| 28 | Measure and integration |
| 30 | Complex analysis |
| 31 | Potential theory |
| 32 | Several complex variables and analytic spaces |
| 33 | Special functions |
| 34 | Ordinary differential equations |
| 35 | Partial differential equations |
| 37 | Dynamical systems and ergodic theory |
| 39 | Difference and functional equations |
| 40 | Sequences, series, summability |
| 41 | Approximations and expansions |
| 42 | Harmonic analysis |
| 43 | Abstract harmonic analysis |
| 44 | Integral transforms, operational calculus |
| 45 | Integral equations |
| 46 | Functional analysis |
| 47 | Operator theory |
| 49 | Calculus of variations and optimal control |
| 51 | Geometry |
| 52 | Convex and discrete geometry |
| 53 | Differential geometry |
| 54 | General topology |
| 55 | Algebraic topology |
| 57 | Manifolds and cell complexes |
| 58 | Global analysis, analysis on manifolds |
| 60 | Probability theory and stochastic processes |
| 62 | Statistics |
| 65 | Numerical analysis |
| 68 | Computer science |
| 70 | Mechanics of particles and systems |
| 74 | Mechanics of deformable solids |
| 76 | Fluid mechanics |
| 78 | Optics, electromagnetic theory |
| 80 | Classical thermodynamics, heat transfer |
| 81 | Quantum theory |
| 82 | Statistical mechanics, structure of matter |
| 83 | Relativity and gravitational theory |
| 85 | Astronomy and astrophysics |
| 86 | Geophysics |
| 90 | Operations research, mathematical programming |
| 91 | Game theory, economics, social and behavioral sciences |
| 92 | Biology and other natural sciences |
| 93 | Systems theory; control |
| 94 | Information and communication; circuits |
| 97 | Mathematics education |
00. General MathematicsLanguage, structure, methodology, and cross-cutting tools that apply across all branches of mathematics.
01. History and BiographyHow mathematical ideas emerged, stabilized, and influenced later work, from prehistory to the modern era.
03. Logic and FoundationsFormal logic, set theory, computability, and the foundations of mathematics treated as a formal system.
05. CombinatoricsCounting, arrangement, structure, and extremal behavior of finite and discrete systems.
06. Order and LatticesPartially ordered sets, lattices, and algebraic systems equipped with order relations.
08. General Algebraic SystemsOperations, identities, and structures as a unifying framework for all algebraic theories.
11. Number TheoryIntegers, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods.
12. Field Theory and PolynomialsThis volume studies fields, polynomials, and algebraic extensions.
13. Commutative AlgebraThis volume studies commutative rings, ideals, modules, and their structural properties.
14. Algebraic GeometryThis volume studies geometric objects defined by polynomial equations.
15. Linear and Multilinear Algebra; Matrix TheoryThis volume develops vector spaces, linear maps, matrices, and multilinear structures.
16. Associative Rings and AlgebrasThis volume studies rings and algebras with associative multiplication, without requiring commutativity.
17. Non-Associative Rings and AlgebrasThis volume studies algebraic systems where associativity does not hold in general.
18. Category Theory; Homological AlgebraThis volume develops category theory as a unifying language and homological algebra as a computational framework for algebraic structures.
19. K-TheoryThis volume studies algebraic and topological K-theory, focusing on invariants derived from vector bundles, modules, and operator algebras.
20. Group Theory and GeneralizationsThis volume studies groups as algebraic structures encoding symmetry.
22. Topological Groups, Lie GroupsThis volume studies groups equipped with topology and smooth structure.
26. Real FunctionsThis volume studies functions of real variables with an emphasis on limits, continuity, differentiation, integration, and fine properties of...
28. Measure and IntegrationThis volume develops measure theory and integration in a general setting.
30. Complex AnalysisThis volume studies functions of a complex variable.
31. Potential TheoryThis volume studies harmonic, subharmonic, and superharmonic functions, along with potentials and their applications to analysis, geometry, and...
32. Several Complex Variables and Analytic SpacesThis volume studies functions of several complex variables, complex manifolds, and analytic spaces.
33. Special FunctionsThis volume studies classical and modern special functions arising as solutions to differential equations, integral transforms, and representation...
34. Ordinary Differential EquationsThis volume studies differential equations involving functions of a single variable.
35. Partial Differential EquationsThis volume studies equations involving partial derivatives of functions in several variables.
37. Dynamical Systems and Ergodic TheoryThis volume studies systems that evolve over time, focusing on long-term behavior, stability, and statistical properties.
39. Difference and Functional EquationsThis volume studies equations defined by discrete steps and functional relations.
40. Sequences, Series, SummabilityThis volume studies convergence, divergence, and summation methods for sequences and series.
41. Approximations and ExpansionsThis volume studies approximation of functions and data by simpler objects such as polynomials, splines, and rational functions.
42. Harmonic AnalysisThis volume studies representation of functions via oscillatory components such as Fourier series and transforms.
43. Abstract Harmonic AnalysisThis volume extends harmonic analysis to general locally compact groups.
44. Integral Transforms, Operational CalculusThis volume studies integral transforms as tools for solving equations, analyzing signals, and transforming problems into more tractable forms.
45. Integral EquationsThis volume studies equations where the unknown function appears under an integral.
46. Functional AnalysisThis volume studies infinite-dimensional vector spaces and linear operators.
47. Operator TheoryThis volume studies linear operators on Banach and Hilbert spaces, with emphasis on spectral properties, structure, and applications in analysis and...
49. Calculus of Variations and Optimal Control; OptimizationThis volume studies optimization of functionals and systems.
51. GeometryThis volume studies geometric structures, transformations, and invariants.
52. Convex and Discrete GeometryThis volume studies convex sets, polytopes, and discrete geometric structures.
53. Differential GeometryThis volume studies smooth geometric structures using calculus.
54. General TopologyThis volume studies topological spaces and continuous structures in full generality.
55. Algebraic TopologyThis volume studies topological spaces through algebraic invariants.
57. Manifolds and Cell ComplexesThis volume studies manifolds and combinatorial models such as CW complexes.
58. Global Analysis, Analysis on ManifoldsThis volume studies analysis on manifolds, combining differential geometry, functional analysis, and partial differential equations.
60. Probability Theory and Stochastic ProcessesThis volume develops probability theory on measure-theoretic foundations and studies stochastic processes.
62. StatisticsThis volume studies statistical inference, estimation, and data analysis.
65. Numerical AnalysisThis volume studies algorithms for approximating mathematical problems.
68. Computer ScienceThis volume studies theoretical and practical foundations of computation.
70. Mechanics of Particles and SystemsThis volume develops classical mechanics using analytical methods.
74. Mechanics of Deformable SolidsThis volume studies the behavior of solid materials under deformation.
76. Fluid MechanicsThis volume studies the motion of fluids and the forces acting on them.
78. Optics, Electromagnetic TheoryThis volume studies light, electromagnetic fields, and wave propagation.
80. Classical Thermodynamics, Heat TransferThis volume studies macroscopic energy, heat, and thermodynamic systems.
81. Quantum TheoryThis volume develops quantum mechanics and its mathematical structure.
82. Statistical Mechanics, Structure of MatterThis volume develops statistical descriptions of many-particle systems.
83. Relativity and Gravitational TheoryThis volume studies spacetime structure, relativistic physics, and gravitation.
85. Astronomy and AstrophysicsThis volume studies celestial objects, their dynamics, and the physical processes governing the universe.
86. GeophysicsThis volume studies the physical processes of the Earth, including its structure, dynamics, and fields.
90. Operations Research, Mathematical ProgrammingThis volume studies decision-making under constraints using mathematical models.
91. Game Theory, Economics, Social and Behavioral SciencesThis volume studies strategic interaction, economic systems, and quantitative models of behavior.
92. Biology and Other Natural SciencesThis volume develops mathematical models for biological and natural systems.
93. Systems Theory; ControlThis volume studies dynamical systems with inputs and outputs.
94. Information and Communication; CircuitsThis volume studies information theory, communication systems, and circuit models.
97. Mathematics EducationThis volume studies the theory and practice of teaching and learning mathematics.