Three Centuries of Discovery

Huygenstwo pendulum clocks on one beam fall into step — synchrony, first noticed

Newtonuniversal gravitation — and the two-body ellipse, solved exactly

de Moivrethe first normal curve, as the limit of many coin flips

d'Alembertsolves the vibrating-string wave equation

Bernoullithe first mathematical epidemic model — weighing smallpox inoculation against the disease

Bayesbelief updates as posterior ∝ likelihood × prior — the rule the filter runs

Lagrangethe equilateral three-body solution and the L-points

Chladnibows a sand-strewn metal plate and the grains leap off the moving regions to settle on the still nodal lines — the first images of a vibration’s modes

Youngthe original two-slit experiment — light interferes, so light is a wave

Gaussleast squares to track Ceres from a few noisy sightings — estimation is born

Laplacethe Central Limit Theorem in its general form

Fourierclaims any periodic signal is a sum of sines (Théorie analytique de la chaleur)

Navierwrites the momentum equation — viscosity and all

Scott Russellobserves a solitary wave of translation on the Union Canal near Edinburgh — a heap of water that travels a mile without changing shape

Stokesputs it on rigorous footing: the modern Navier–Stokes form

Cauchymethod of steepest descent — follow the negative gradient downhill, the optimiser still in use

Wilbrahamfirst spots the overshoot at a jump — then it is forgotten for fifty years

Clausiusnames entropy and states the second law of thermodynamics

Maxwellproposes a tiny "demon" that sorts fast from slow molecules — apparently defeating the second law without doing work

Boltzmannthe H-theorem and S = k ln Ω — irreversibility from counting

Loschmidtthe reversibility objection: symmetric laws, an asymmetric world

Reynoldshis number, and the dye experiment that catches laminar flow breaking up

Michelson–Morleyfind no aether — light's speed refuses to add up

Poincaréno closed form for three bodies — the first sight of chaos

Poincarérecurrence: a closed system must eventually return near its start

Lyapunovhis stability theory gives the exponent λ that measures the stretch

Korteweg & de Vriesderive the KdV equation for shallow water waves and find its exact soliton solution, vindicating Scott Russell 60 years later

Gibbsexplains it: the ripple narrows but homes in on ≈8.95%, never vanishing

Bacheliermodels market prices as a random walk — finance meets diffusion, the first time-series

EinsteinBrownian motion ties the walk to atoms and Avogadro's number

Einsteinspecial relativity: c is invariant, and space and time mix

Markovthe Markov chain — a process whose next step depends only on the present, with a long-run equilibrium

Minkowskispacetime — the geometry in which the light cone is absolute

Langevinthe Langevin equation — Newton plus a random force; the stochastic differential equation that underlies every diffusion here

Haarthe first wavelet — a square step that splits a signal by scale

WeylWeyl’s law — the count of eigenfrequencies below a bound grows with the drum’s area, so the spectrum does betray the size, if not the shape

Perron & Frobeniusa positive matrix has one largest eigenvalue with an all-positive eigenvector — the unique, well-defined rank

Einsteingeneral relativity: matter curves spacetime, and light follows the curve

Whittakerthe interpolation formula that rebuilds a signal from its samples

Schwarzschildthe first exact solution — and with it, the event horizon

Julia & Fatouthe dynamics of z² + c — decades before anyone could see it

Eddingtonthe eclipse measures the bending at twice Newton's value — Einstein, famous overnight

Bosenew statistics for light quanta — particles that bunch together

de Brogliematter has a wavelength too — so particles can interfere

Einsteinextends Bose statistics to atoms and predicts the condensate

Isingsolves the 1D chain — and finds no phase transition (the model nearly dies here)

Lotkaderives the oscillating predator–prey equations in Elements of Physical Biology — a population read as a chemical-style dynamical system

Schrödingerthe wave equation — the whole future of ψ

Born|ψ|² is probability: the cloud itself (Nobel 1954)

Volterraexplains why the WWI pause in Adriatic fishing raised the shark fraction — the same equations, from his son-in-law fish-market data

Hundfirst spots tunnelling, in the splitting of molecular spectra

Madelungrewrites ψ as a fluid — a density and a flowing phase

Kermack & McKendrickthe SIR model and the epidemic threshold: an outbreak needs R₀ > 1 to take off

Gamowexplains alpha decay by tunnelling — radioactivity as a quantum leak

Nyquistfixes the rate: sample above twice the top frequency

Hubblegalaxies recede at a speed proportional to their distance — the universe is expanding

Szilárdresolves the demon with a one-molecule engine: one measurement yields kT ln 2 of work — and costs one bit of memory

Kotelnikovproves the sampling theorem in full

Zwickythe Coma cluster moves far too fast for its visible mass — most of the gravitating matter is unseen

Gausetests predator–prey in test tubes of Paramecium and Didinium — the predator eats all prey then starves; coexistence needs a refuge

Einstein, Podolsky & Rosenargue QM must be incomplete — "spooky action at a distance" can't be real

Whitneythe embedding theorem — any smooth d-manifold sits faithfully inside ℝ^(2d+1)

Landauthe general theory of phase transitions and the order parameter

Londonties superfluid helium to Bose–Einstein condensation

Onsagersolves the 2D Ising model exactly — T_c and the critical exponents, by hand

Gutenberg & Richterearthquake magnitudes follow a power law — no characteristic size, decades before anyone could say why

Itôstochastic calculus: the lemma that integrates dS exactly into a log-normal

von Neumann & MorgensternTheory of Games and Economic Behavior founds game theory — rational players, payoff matrices and the minimax theorem

Gabortiles time–frequency into "logons" and finds the joint-resolution limit

Shannon'A Mathematical Theory of Communication' — entropy, capacity, the whole field

von Neumann & Ulaminvent cellular automata — a self-reproducing machine on a grid

Onsagercirculation is quantised: vortices carry integer charge

Shannonmakes the sampling theorem a cornerstone of digital signal processing

Ulam · Metropolisthe Monte Carlo method — answer by sampling when the formula is hard

Hebb'neurons that fire together wire together' — memory as the strengthening of a connection

Hammingthe first error-correcting code — parity that names the flipped bit

Nashthe Nash equilibrium — every finite game has a stable strategy profile no player can beat by deviating alone

Belousovdiscovers a chemical reaction that oscillates in colour instead of settling to equilibrium — rejected as impossible by the journals, and ignored for a decade

Turing'The Chemical Basis of Morphogenesis' — diffusion can make pattern, not just smooth it

Metropolis, Rosenbluth & Tellerthe Metropolis algorithm — sample a Boltzmann distribution by accepting moves with probability min(1, e^−ΔE/T)

Esakithe tunnel diode puts the effect to work (Nobel 1973)

Broadbent & Hammersleyfounding paper on percolation — originally motivated by fluid flow through random media

Bellmandynamic programming and the principle of optimality — the value of a state is its best reward plus the discounted value of where you land; also names the curse of dimensionality

Rosenblattthe perceptron: a learning machine of weighted neurons — but a single layer cannot solve XOR

Hollingthe functional response — a predator saturates because prey take time to handle (type II); the realism that lets the model cycle and crash

Reed & Solomoncodes that armour CDs, QR codes and deep-space links

Kalmanthe recursive filter: optimal estimation of a linear system, one step at a time

Jönssonruns the double slit with single electrons, one at a time

Landauershows that erasing one bit of information dissipates at least kT ln 2 as heat — information is physical

FitzHugh & Nagumoreduce the Hodgkin–Huxley nerve impulse to two variables — a fast excitable voltage and a slow recovery — the minimal model of a firing, refractory medium

Lorenzthree equations for convection — the butterfly effect, chaos found by accident from a rounded printout

Rosenzweig & MacArthura graphical predator–prey model with logistic prey and saturating predation — isoclines that predict a stable point or a limit cycle

Sharkovskiiorders the periods: a map with a 3-cycle must have cycles of every length

Bellturns "is the world local?" into an inequality you can actually measure

Zhabotinskyrevives Belousov’s reaction and finds it makes travelling and rotating waves in a dish — chemistry that propagates like a nerve impulse or a heartbeat

Penrosecollapse to a singularity is inevitable — black holes are a robust prediction of relativity, not a fluke (Nobel 2020)

Cooley & Tukeythe FFT makes the transform cheap — now inside MP3, JPEG, every analyser

Kirkpatrick et al.Monte Carlo computation of p_c for the square lattice ≈ 0.5927

Samuelsongeometric (not arithmetic) Brownian motion for prices — they can never go negative

Zabusky & Kruskalrediscover solitons on a computer, observe that two pulses pass through each other unchanged — and coin the name "soliton"

Kac“Can one hear the shape of a drum?” — asks whether two differently-shaped membranes could ever share an identical spectrum of tones

Winfreebiological oscillators sync once the coupling beats the spread

Mandelbrot'How Long Is the Coast of Britain?' — fractional dimension enters science: the length depends on the ruler

Viterbithe Viterbi algorithm — dynamic programming recovers the single most-likely hidden path through a noisy sequence, exactly, in one backward sweep

Venezianohis amplitude for the strong force secretly describes a string

Clauser, Horne, Shimony & Holtthe experimentally testable form of Bell's bound, S ≤ 2

Apollothe Kalman filter flies the guidance computer to the Moon and back

Nambu, Nielsen & Susskindread it literally: the constituents are vibrating strings

Conwaythe Game of Life makes cellular automata famous

Gosperfinds the glider gun — Life grows without bound, so it can store and move information

Zel'dovichthe approximation that lets gravity be run by hand — matter collapses first into sheets (pancakes), then filaments, then knots

Baum, Petrie, Soules & Weissthe hidden Markov model with forward–backward and Baum–Welch — infer the hidden states, and the chain itself, from the observations alone

Wilsonthe renormalisation group explains universality (Nobel 1982)

Hafele–Keatingflying atomic clocks confirm time dilation directly

Rosenzweigthe paradox of enrichment — raising the prey carrying capacity destabilises the equilibrium into ever-wider cycles that can drive extinction

Gierer & Meinhardtname the design rule: short-range activation, long-range inhibition

Mayasks whether a large complex system is stable and finds it usually is not — richly connected ecosystems tend to be less stable, not more

Winfreeshows the rotating chemical spiral turns about a phase singularity — a point where phase is undefined — and ties the same geometry to cardiac fibrillation

Black · Scholes · Mertonthe same log-normal, priced: option value as a discounted forecast

Maynard Smith & Pricethe evolutionarily stable strategy — game theory carried into biology, where fitness rather than reason selects the move

Littlecasts a neural network as a spin system — persistent firing states are the stored memories

Li & Yorkecoin the word — “Period Three Implies Chaos”

Kuramotothe exactly solvable synchronization model — order parameter r and the critical K_c

Robert Maythe logistic map as a toy ecology (Nature) — a simple law, an uncomputable fate

Feigenbaumfinds the doubling ratio δ ≈ 4.6692 is universal — on a pocket calculator

Kaplan–Yorkeconjecture links the Lyapunov exponents to the attractor's fractal dimension

Benettinan algorithm to measure λ from a shadow orbit — the method the rack uses live

Tsirelsonproves quantum mechanics can violate CHSH only up to 2√2

Mandelbrotcomputes the set at IBM and reveals the shape

Peeblesthe gravitational theory of how faint early-universe ripples grow into the cosmic web (Nobel 2019)

Packard, Crutchfield, Farmer & Shaw“Geometry from a Time Series” — reconstruct an attractor from a single signal’s own delayed copies

Binnig & Rohrerthe scanning tunnelling microscope feels single atoms (Nobel 1986)

Witten & Sanderdiffusion-limited aggregation — random walkers stick on contact into a branching fractal of dimension ≈ 1.71

Takensproves it: delay-coordinate embedding is generically diffeomorphic to the true attractor — same invariants

Axelrod & Hamiltonthe iterated tournament — Tit-for-Tat beats every cleverer rule; cooperation evolves among selfish agents that meet again

Benzi, Sutera & Vulpianiintroduce stochastic resonance to explain the 100,000-year ice-age cycle: orbital forcing alone is too weak — noise from climate fluctuations makes it detectable

Sparrowthe definitive study of the Lorenz equations

Aspectswitches the analysers in flight — closes the locality loophole, S ≈ 2.7

Nienhuisexact critical exponents β = 5/36, ν = 4/3 via Coulomb gas / conformal field theory

Hopfielda recurrent net whose energy only falls, so memories are the basins of attraction it rolls into

Berlekamp, Conway & Guy'Winning Ways' proves Life is Turing-complete — a universal computer built from one rule

Douady & Hubbardprove M is connected, and give it his name

Bennettproves the demon cannot violate the second law: the erasure of its memory ledger, not the measurement, is the thermodynamic cost

Andersonthe reverse-time SDE — a diffusion run backward is itself a diffusion, driven by the score of the density; the theorem generative models would later stand on

Gray & Scottthe autocatalytic reaction–diffusion model this engine runs

Wolframsorts the 256 elementary rules into four behaviours

Grassberger & Procacciathe correlation dimension — read a strange attractor’s fractal dimension straight off the reconstructed cloud

Kirkpatrick, Gelatt & VecchiOptimization by Simulated Annealing — melt a hard problem, then cool it slowly through Metropolis moves into its best state

Morlet & Grossmannformalise the continuous wavelet transform, out of seismic traces

Brady & Ballcopper electrodeposits grow as DLA fractals — the model caught in the lab, the same dimension measured in metal

Amit, Gutfreund & Sompolinskythe spin-glass calculation: the net stores ≈0.138N patterns before the basins dissolve

Černýindependently applies thermodynamic annealing to the travelling-salesman problem — the method arrives twice at once

Rumelhart, Hinton & Williamsbackpropagation trains hidden layers efficiently — the multilayer network finally learns

ReynoldsBoids: three local rules — align, cohere, separate

Bak, Tang & Wiesenfeldself-organized criticality: a sandpile tunes itself to the critical point and emits 1/f noise — no parameter set by hand

Daubechiescompactly-supported orthonormal wavelets — the basis inside JPEG 2000

Suttontemporal-difference learning — update a prediction from a later, better prediction, with no model of the world; the bracket that becomes Q-learning

Laskarthe inner Solar System is chaotic, with a ~5-million-year horizon

Mallatmultiresolution analysis + the fast wavelet transform

Hart, Sandin & Kauffmandistance-estimated ray tracing — march a ray by a safe lower bound on the distance to the surface, which is what makes a 3D escape-time fractal renderable at all

WatkinsQ-learning — a model-free, off-policy rule that provably converges to the optimal action-values just by trying, watching and updating

Hamiltonthe Markov-switching model carries hidden-state inference into economics — reading booms and recessions as latent regimes behind the noise of GDP and returns

De Kepperfirst chemical Turing patterns seen in the lab (the CIMA reaction)

Dharthe abelian sandpile: topplings commute, the recurrent states form a group, and the identity is a fractal

Shishikuraproves the Mandelbrot boundary has Hausdorff dimension 2

Nowak & Mayspatial games — cooperators and defectors on a grid form ever-shifting fractal patterns; structure alone sustains cooperation

TesauroTD-Gammon reaches near-expert backgammon by playing millions of games against itself — temporal-difference learning, scaled up, beats hand-tuned programs

Gordon, Webb & Wolpertanswers Kac — no: they build two different polygonal drums with exactly the same spectrum. You can hear the area and perimeter, but not the shape

Berrouturbo codes reach within a whisker of the Shannon limit

Watts & Strogatz · Barabásipercolation transitions in real-world networks: internet, epidemics, markets

Moss, Pierson & O'Gormandemonstrate stochastic resonance in biological neurons — sensory systems appear to operate near the optimal noise level for signal detection

Englert–Greenberger–Yasinpin down the exact duality relation V² + D² ≤ 1

Cornell, Wieman & Ketterlethe first gaseous BEC, at JILA and MIT (Nobel 2001)

Vicsekthe minimal model — collective motion as a true phase transition

Watts & Strogatzsmall-world networks — a few long-range links collapse the distance a contagion must travel

Page & BrinPageRank — rank the web by its link structure alone; a random surfer’s equilibrium, and the founding of Google

Gammaitoni, Hänggi, Jung & Marchesonipublish the landmark review of stochastic resonance, establishing it as a universal phenomenon in nonlinear noisy systems

Jos Stam“Stable Fluids” — an unconditionally stable solver; real-time fluids in a browser

Pastor-Satorras & Vespignanion scale-free networks the epidemic threshold can vanish — hubs change everything

Couzinmaps the switch between swarm, torus and aligned flock

Cookproves Rule 110 is Turing-complete — a universal computer

Strogatzthe Millennium Bridge wobble: pedestrians lock step with the sway

Ballerinireal murmurations: each bird tracks ~7 neighbours, by rank not distance

White & Nylanderthe Mandelbulb — take the nth power of a point in spherical coordinates and z → zⁿ + c becomes a 3D fractal with genuine surface detail

Krizhevsky, Sutskever & HintonAlexNet: the same gradient descent, scaled to GPUs, wins ImageNet and ignites deep learning

Sugihara et al.convergent cross mapping — “shadow manifolds” turn reconstruction into a test for causation between time series

DeepMindthe Deep Q-Network learns to play 49 Atari games from raw pixels and the score alone, at human level — Q-learning with a deep net as the value function

Sohl-Dickstein et al.deep unsupervised learning by nonequilibrium thermodynamics — destroy data with a slow diffusion, then train a network to reverse each step

DeepMindAlphaGo defeats Lee Sedol at Go — self-play reinforcement learning plus tree search solves a game long thought a decade away

Event Horizon Telescopephotographs a black hole's shadow for the first time

Song & Ermonscore-based generative modelling — learn the gradient of the log-density and sample by following it with Langevin dynamics

Ho, Jain & Abbeeldenoising diffusion probabilistic models — the clean recipe that made diffusion the engine behind modern image generators

Song et al.unifies diffusion as a reverse-time SDE with a deterministic probability-flow ODE twin — one framework for every sampler

Aspect, Clauser & Zeilingerthe Nobel for the experiments proving the world is not locally real

Hopfield & Hintonthe Nobel Prize in Physics for the physics of memory and learning in neural networks