Sources

Every dismissal and every application, field by field.

Links and citations, in one place, matched to each field's own page.

Mathematical logic & foundations (MSC 03)

Poincaré, essays collected in Science et méthode, c. 1905–06, verified against the primary French text.
Model checking (1981): https://en.wikipedia.org/wiki/Model_checking
Formally verified software (seL4) (2009): https://en.wikipedia.org/wiki/L4_microkernel_family#seL4

Combinatorics (MSC 05)

Design of experiments (1925): https://en.wikipedia.org/wiki/Design_of_experiments
Reed–Solomon codes (1960): https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction

Order, lattices & Boolean algebra (MSC 06/08)

Digital circuit design (1937): https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuits

Number theory (MSC 11)

Hardy, A Mathematician's Apology, CUP 1940, §21/§25/§28, verified against the primary text.
RSA cryptography (1977): https://dl.acm.org/doi/10.1145/359340.359342

Field theory & Galois theory (MSC 12)

Reed–Solomon codes (1960): https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction

Commutative rings & algebras (MSC 13)

McEliece cryptosystem (1978): https://en.wikipedia.org/wiki/McEliece_cryptosystem
NIST post-quantum standards (2024): https://csrc.nist.gov/pubs/fips/203/final

Algebraic geometry (MSC 14)

Koblitz et al., IACR ePrint 2008/390.
Elliptic-curve cryptography (1985): https://en.wikipedia.org/wiki/Elliptic-curve_cryptography

Linear & multilinear algebra; matrix theory (MSC 15)

Matrix mechanics (1925): https://en.wikipedia.org/wiki/Matrix_mechanics
Google PageRank (1998): https://en.wikipedia.org/wiki/PageRank

Associative rings & algebras (quaternions) (MSC 16)

Letter to Robert Baldwin Hayward, 1892; standard citation Silvanus Thompson's 1910 biography of Kelvin.
3D computer graphics (SLERP) (1985): https://en.wikipedia.org/wiki/Slerp
Gimbal-lock-free attitude control (1985): https://en.wikipedia.org/wiki/Gimbal_lock

Lie algebras & nonassociative algebra (MSC 17)

The Eightfold Way (1961): https://en.wikipedia.org/wiki/Eightfold_way_(physics)

Category theory & homological algebra (MSC 18)

Attribution via Rotman (Bull. AMS 33:4, 1996) and McLarty (Br. J. Philos. Sci. 41, 1990); no primary quote from Steenrod himself has surfaced.
Monads in functional programming (1989): https://en.wikipedia.org/wiki/Monad_(functional_programming)

K-theory (MSC 19)

Classification of topological phases of matter (2009): https://en.wikipedia.org/wiki/Topological_insulator

Potential theory (MSC 31)

Electrostatics & gravitational modeling (1828): https://en.wikipedia.org/wiki/Potential_theory

Several complex variables (MSC 32)

Multidimensional systems engineering (1975): https://en.wikipedia.org/wiki/Multidimensional_filter_design

Special functions (MSC 33)

Elliptic-curve cryptography (1985): https://en.wikipedia.org/wiki/Elliptic-curve_cryptography

Ordinary differential equations (MSC 34)

Orbital mechanics (1687): https://en.wikipedia.org/wiki/Ordinary_differential_equation

Partial differential equations (MSC 35)

Finite element method (1956): https://en.wikipedia.org/wiki/Finite_element_method

Dynamical systems & ergodic theory (MSC 37)

Chaos theory & weather-forecast limits (1963): https://en.wikipedia.org/wiki/Chaos_theory

Classical analysis: series & summability (MSC 40/41)

Abel's Oeuvres complètes; Wikipedia's “Borel summation” article for the Mittag-Leffler line.
Critical exponents in condensed matter (1977): https://en.wikipedia.org/wiki/Renormalization_group

Fourier analysis (MSC 42)

Tide-predicting machine (1872): https://en.wikipedia.org/wiki/Tide-predicting_machine

Abstract harmonic analysis (MSC 43)

Fast SO(3) Fourier transforms (2000): https://en.wikipedia.org/wiki/Wigner_D-matrix

Integral transforms & integral equations (MSC 44/45)

MacTutor History of Mathematics.
Feedback amplifier design (1927): https://en.wikipedia.org/wiki/Hendrik_Wade_Bode
Actuarial calculations (1904): https://en.wikipedia.org/wiki/Fredholm_integral_equation

Functional analysis & operator theory (MSC 46/47)

Widely attributed via Lynn Steen, American Mathematical Monthly 80(4), 1973; primary provenance unconfirmed.
Quantum mechanics (1932): https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics

Calculus of variations & optimal control (MSC 49)

Rocket trajectory optimization (1956): https://en.wikipedia.org/wiki/Pontryagin%27s_maximum_principle

Geometry: classical, convex & discrete (MSC 51/52)

Coding theory (Leech lattice) (1965): https://en.wikipedia.org/wiki/Leech_lattice
Lattice-based post-quantum cryptography (2005): https://en.wikipedia.org/wiki/Lattice-based_cryptography

Differential geometry (MSC 53)

Letter to Bessel, 1829.
General Relativity (1915): https://en.wikipedia.org/wiki/General_relativity
GPS (1977): https://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System

General topology (MSC 54)

General equilibrium economics (1954): https://en.wikipedia.org/wiki/Arrow%E2%80%93Debreu_model

Algebraic topology (MSC 55)

Topological data analysis (2002): https://en.wikipedia.org/wiki/Topological_data_analysis
Breast-cancer subtype discovery (2011): https://doi.org/10.1073/pnas.1102826108

Manifolds & low-dimensional topology (knot theory) (MSC 57/58)

Kelvin, “On Vortex Atoms,” Proc. Royal Society of Edinburgh, 1867.
DNA topology (1990): https://en.wikipedia.org/wiki/Topoisomerase
Topological quantum computing (1997): https://en.wikipedia.org/wiki/Topological_quantum_computer

Probability theory (MSC 60)

Actuarial science & statistical mechanics (1654): https://en.wikipedia.org/wiki/Probability_theory
Black–Scholes options pricing (1973): https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

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