Maths
Notes on mathematics, organized by MSC classification.
Maths
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]({{< relref "maths/00/_index.md" >}}) | General Mathematics | Language, structure, methodology, and cross-cutting tools across all branches |
| [01]({{< relref "maths/01/_index.md" >}}) | History and Biography | How mathematical ideas emerged, stabilized, and influenced later work |
| [03]({{< relref "maths/03/_index.md" >}}) | Logic and Foundations | Formal logic, set theory, computability, proof theory |
| [05]({{< relref "maths/05/_index.md" >}}) | Combinatorics | Counting, arrangement, structure, and extremal behavior of finite systems |
| [06]({{< relref "maths/06/_index.md" >}}) | Order and Lattices | Partially ordered sets, lattices, and ordered algebraic structures |
| [08]({{< relref "maths/08/_index.md" >}}) | General Algebraic Systems | Operations and identities as a unifying framework for all algebraic theories |
| [11]({{< relref "maths/11/_index.md" >}}) | Number Theory | Integers, primes, modular arithmetic, Diophantine equations |
Full MSC index
Numbers without a link are planned but not started yet.
| # | Area |
|---|---|
| [00]({{< relref "maths/00/_index.md" >}}) | General and overarching topics |
| [01]({{< relref "maths/01/_index.md" >}}) | History and biography |
| [03]({{< relref "maths/03/_index.md" >}}) | Mathematical logic and foundations |
| [05]({{< relref "maths/05/_index.md" >}}) | Combinatorics |
| [06]({{< relref "maths/06/_index.md" >}}) | Order, lattices, ordered algebraic structures |
| [08]({{< relref "maths/08/_index.md" >}}) | General algebraic systems |
| [11]({{< relref "maths/11/_index.md" >}}) | 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 Mathematics
Language, structure, methodology, and cross-cutting tools that apply across all branches of mathematics.
01. History and Biography
How mathematical ideas emerged, stabilized, and influenced later work, from prehistory to the modern era.
03. Logic and Foundations
Formal logic, set theory, computability, and the foundations of mathematics treated as a formal system.
05. Combinatorics
Counting, arrangement, structure, and extremal behavior of finite and discrete systems.
06. Order and Lattices
Partially ordered sets, lattices, and algebraic systems equipped with order relations.
08. General Algebraic Systems
Operations, identities, and structures as a unifying framework for all algebraic theories.
11. Number Theory
Integers, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods.
12. Field Theory and Polynomials
This volume studies fields, polynomials, and algebraic extensions.
13. Commutative Algebra
This volume studies commutative rings, ideals, modules, and their structural properties.
14. Algebraic Geometry
This volume studies geometric objects defined by polynomial equations.
15. Linear and Multilinear Algebra; Matrix Theory
This volume develops vector spaces, linear maps, matrices, and multilinear structures.
16. Associative Rings and Algebras
This volume studies rings and algebras with associative multiplication, without requiring commutativity.
17. Non-Associative Rings and Algebras
This volume studies algebraic systems where associativity does not hold in general.
18. Category Theory; Homological Algebra
This volume develops category theory as a unifying language and homological algebra as a computational framework for algebraic structures.
19. K-Theory
This volume studies algebraic and topological K-theory, focusing on invariants derived from vector bundles, modules, and operator algebras.
20. Group Theory and Generalizations
This volume studies groups as algebraic structures encoding symmetry.
22. Topological Groups, Lie Groups
This volume studies groups equipped with topology and smooth structure.
26. Real Functions
This volume studies functions of real variables with an emphasis on limits, continuity, differentiation, integration, and fine properties of...
28. Measure and Integration
This volume develops measure theory and integration in a general setting.
31. Potential Theory
This volume studies harmonic, subharmonic, and superharmonic functions, along with potentials and their applications to analysis, geometry, and...
32. Several Complex Variables and Analytic Spaces
This volume studies functions of several complex variables, complex manifolds, and analytic spaces.
33. Special Functions
This volume studies classical and modern special functions arising as solutions to differential equations, integral transforms, and representation...
34. Ordinary Differential Equations
This volume studies differential equations involving functions of a single variable.
35. Partial Differential Equations
This volume studies equations involving partial derivatives of functions in several variables.
37. Dynamical Systems and Ergodic Theory
This volume studies systems that evolve over time, focusing on long-term behavior, stability, and statistical properties.
39. Difference and Functional Equations
This volume studies equations defined by discrete steps and functional relations.
40. Sequences, Series, Summability
This volume studies convergence, divergence, and summation methods for sequences and series.
41. Approximations and Expansions
This volume studies approximation of functions and data by simpler objects such as polynomials, splines, and rational functions.
42. Harmonic Analysis
This volume studies representation of functions via oscillatory components such as Fourier series and transforms.
43. Abstract Harmonic Analysis
This volume extends harmonic analysis to general locally compact groups.
44. Integral Transforms, Operational Calculus
This volume studies integral transforms as tools for solving equations, analyzing signals, and transforming problems into more tractable forms.
45. Integral Equations
This volume studies equations where the unknown function appears under an integral.
46. Functional Analysis
This volume studies infinite-dimensional vector spaces and linear operators.
47. Operator Theory
This 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; Optimization
This volume studies optimization of functionals and systems.
52. Convex and Discrete Geometry
This volume studies convex sets, polytopes, and discrete geometric structures.
54. General Topology
This volume studies topological spaces and continuous structures in full generality.
57. Manifolds and Cell Complexes
This volume studies manifolds and combinatorial models such as CW complexes.
58. Global Analysis, Analysis on Manifolds
This volume studies analysis on manifolds, combining differential geometry, functional analysis, and partial differential equations.
60. Probability Theory and Stochastic Processes
This volume develops probability theory on measure-theoretic foundations and studies stochastic processes.
65. Numerical Analysis
This volume studies algorithms for approximating mathematical problems.
68. Computer Science
This volume studies theoretical and practical foundations of computation.
70. Mechanics of Particles and Systems
This volume develops classical mechanics using analytical methods.
74. Mechanics of Deformable Solids
This volume studies the behavior of solid materials under deformation.
78. Optics, Electromagnetic Theory
This volume studies light, electromagnetic fields, and wave propagation.
80. Classical Thermodynamics, Heat Transfer
This volume studies macroscopic energy, heat, and thermodynamic systems.
82. Statistical Mechanics, Structure of Matter
This volume develops statistical descriptions of many-particle systems.
83. Relativity and Gravitational Theory
This volume studies spacetime structure, relativistic physics, and gravitation.
85. Astronomy and Astrophysics
This volume studies celestial objects, their dynamics, and the physical processes governing the universe.
86. Geophysics
This volume studies the physical processes of the Earth, including its structure, dynamics, and fields.
90. Operations Research, Mathematical Programming
This volume studies decision-making under constraints using mathematical models.
91. Game Theory, Economics, Social and Behavioral Sciences
This volume studies strategic interaction, economic systems, and quantitative models of behavior.
92. Biology and Other Natural Sciences
This volume develops mathematical models for biological and natural systems.