Who is Who in Mathematics

Died in a duel at twenty, having the night before written his mathematical discoveries in a letter. His theory of groups and field extensions answered the 2,000-year-old question of which polynomial equations are solvable by radicals, and founded modern abstract algebra.

Can help you study: Galois theory, its proof that degree-5 equations are not generally solvable by radicals, and the founding of group theory.

First defined abstract group theory as a set with a binary operation satisfying four axioms, introduced matrix algebra as a formal system, and proved the Cayley-Hamilton theorem.

Can help you study: Matrix algebra, abstract group theory, the Cayley-Hamilton theorem, and invariant theory.

The most important woman in the history of mathematics. Transformed algebra from a subject about specific operations to one about abstract structures. Her theorem connecting symmetry to conservation laws underpins all of modern physics.

Can help you study: Noether’s theorem and conservation laws, abstract algebra, ring and ideal theory, and the structural approach to mathematics she invented.

Co-inventor of category theory, which studies mathematical structures and the mappings between them rather than their internal elements. Category theory has become the unifying language of pure mathematics.

Can help you study: Category theory, functors and natural transformations, homological algebra, and why studying maps between structures matters as much as the structures themselves.

Showed that any periodic function can be expressed as an infinite sum of sines and cosines while studying heat flow. Launched harmonic analysis and gave science the Fourier transform, underpinning signal processing and quantum mechanics.

Can help you study: The Fourier series and transform, harmonic analysis, the heat equation, and why representing functions as sums of sines and cosines is so powerful.

Put calculus on a rigorous footing with epsilon-delta definitions of limits and continuity. His complex analysis — the residue theorem, Cauchy’s integral formula — is equally foundational.

Can help you study: Rigorous analysis, epsilon-delta limits, complex analysis, the residue theorem, and the project of making calculus logically watertight.

The father of modern analysis: gave rigorous definitions of uniform convergence and continuity, and constructed a continuous nowhere-differentiable function that shocked contemporaries.

Can help you study: Uniform vs pointwise convergence, continuous nowhere-differentiable functions, and the rigorous foundations of real analysis.

Developed measure theory and the Lebesgue integral, extending integration to a far wider class of functions and providing the foundation of modern probability and functional analysis.

Can help you study: Measure theory, the Lebesgue integral and how it differs from Riemann, and why a more general notion of integration was needed.

Founded functional analysis, defined Banach spaces, proved the Hahn-Banach and open mapping theorems, and gave his name to the Banach-Tarski paradox. His Scottish Café notebooks became legendary.

Can help you study: Banach spaces, the Hahn-Banach theorem, the Banach-Tarski paradox, and the mathematical culture of interwar Lviv.

Introduced Sobolev spaces and weak derivatives, providing the right framework for studying solutions to partial differential equations and making rigorous a wide class of applied problems.

Can help you study: Sobolev spaces, weak derivatives, PDEs and their solutions, and distribution theory.

Provided the rigorous theory of distributions, making the physicist’s Dirac delta function and similar objects mathematically precise. Théorie des distributions is one of the major mathematical works of the twentieth century.

Can help you study: Distribution theory, the rigorous treatment of the Dirac delta, generalised functions, and their applications in analysis.

Greek mathematician of Alexandria whose Arithmetica — thirteen books, six surviving in Greek and four more in Arabic translation — introduced symbolic algebraic notation and a method for solving what are now called Diophantine equations: polynomial equations to be solved in rational or integer values. His single unknown, the arithmos (ς), was a genuine notational innovation: he reduced every problem to one unknown, eliminating all others by substitution. It was in the margin of a Latin translation of the Arithmetica that Fermat wrote his Last Theorem. Almost nothing is known of his life; his approximate dates are inferred from a dedication and an arithmetical riddle in an epigram ascribed to him.

Can help you study: Diophantine equations, algebraic notation, rational solutions to polynomial equations, the Arithmetica, the history of algebra, and the reduction of multi-variable problems to a single unknown.

Greek polymath and Chief Librarian of the Library of Alexandria who calculated the circumference of the Earth to within a few percent, using the angle of a shadow at noon and the known distance between Syene and Alexandria. He also invented the Sieve of Eratosthenes — the oldest algorithm for generating prime numbers, still in use — mapped the known world in systematic coordinates, measured the tilt of the Earth’s axis, and wrote on astronomy, geography, history, and literary criticism. His contemporaries called him ‘Beta’ — second-best in everything — which he took as a compliment: he was competitive across every domain. Eratosthenes went blind in old age and, unable to read, starved himself to death.

Can help you study: The Sieve of Eratosthenes, prime numbers, geodesy, the measurement of the Earth’s circumference, mathematical geography, cross-disciplinary thinking, and the history of the ancient Library of Alexandria.

French lawyer and amateur mathematician who founded modern number theory in the margins of his copy of Diophantus. Fermat’s Last Theorem — that no three positive integers satisfy aⁿ + bⁿ = cⁿ for n > 2 — took 358 years to prove (Wiles, 1995). He also co-invented probability theory (with Pascal), discovered Fermat’s little theorem, and formulated the principle of least time in optics. Toulouse.

Can help you study: Number theory, Fermat’s Last Theorem, Fermat’s little theorem, the principle of least time, probability, and the art of stating theorems without proving them.

Swiss mathematician who produced more mathematics than any other individual in history — over 800 papers, filling nearly 80 volumes. He unified the constants e, i, π, 0, and 1 in a single identity. He invented graph theory, contributed to every branch of mathematics that existed, and continued working after going blind. Basel, St Petersburg, Berlin, St Petersburg again.

Can help you study: Number theory, graph theory, Euler’s identity, the zeta function, combinatorics, mechanics, the calculus of variations, and the principle that mathematics is unlimited in its scope.

The Prince of Mathematics. By the age of twenty-four he had proved the fundamental theorem of algebra, published the Disquisitiones Arithmeticae (the founding text of modern number theory), and computed the orbit of Ceres from three observations. He also invented the method of least squares, the Gaussian distribution, non-Euclidean geometry (which he kept secret), and Gauss’s law. Göttingen, for nearly fifty years.

Can help you study: Number theory, modular arithmetic, quadratic reciprocity, the Gaussian distribution, least squares, differential geometry, and the standard by which mathematical genius is measured.

Indian mathematician who, with almost no formal training, produced results in number theory, infinite series, and continued fractions that astonished the professionals. His letter to Hardy at Cambridge (1913) contained theorems so remarkable that Hardy said they had to be true, because no one could have invented them. He came to England, was elected a Fellow of the Royal Society, and died at thirty-two.

Can help you study: Number theory, partition functions, infinite series, continued fractions, modular forms, the Ramanujan conjecture, and the mystery of mathematical intuition without formal training.

Hungarian mathematician who published over 1,500 papers with over 500 collaborators — more than anyone in history. He had no home, no possessions, and no fixed position. He travelled from mathematician to mathematician with a suitcase, offering prizes for solved problems. His productivity came from collaboration, and the concept of the Erdős number measures how close any mathematician is to having co-authored with him.

Can help you study: Number theory and combinatorics, the Erdős number and collaboration networks, probabilistic methods in combinatorics, graph theory, and the question of what mathematics looks like when done as a collective, itinerant practice.

Hungarian mathematician who published more papers (over 1,500) with more collaborators (over 500) than anyone in history. He had no home, no possessions, and no job. He travelled from mathematician to mathematician, arriving at the door and announcing that his brain was open. He co-invented the probabilistic method, made foundational contributions to combinatorics and number theory, and proved theorems until the day he died at a conference in Warsaw.

Can help you study: Combinatorics, number theory, the probabilistic method, Ramsey theory, graph theory, the Erdős number, and the principle that mathematics is a social activity.

English mathematician who created modern statistics. The correlation coefficient, the chi-squared test, the method of moments, principal component analysis, the histogram. He founded the world’s first university statistics department (UCL, 1911) and edited Biometrika for over thirty years. His contributions to eugenics are a permanent stain; his contributions to statistical method are permanent foundations.

Can help you study: Correlation, the chi-squared test, principal component analysis, biometrics, the method of moments, and the foundations of statistical methodology.

English statistician and geneticist who invented the analysis of variance, the design of experiments, maximum likelihood estimation, and the randomised controlled trial. His Statistical Methods for Research Workers (1925) and The Design of Experiments (1935) defined how science does statistics. He also reconciled Mendelian genetics with Darwinian evolution. Rothamsted, Cambridge, Adelaide.

Can help you study: Experimental design, ANOVA, maximum likelihood, significance testing, the lady tasting tea, randomisation, and the statistical foundations of modern science.

Soviet mathematician who axiomatised probability theory (1933), placing it on the same rigorous foundation as the rest of mathematics. He also made foundational contributions to turbulence theory, algorithmic complexity, and dynamical systems. One of the most versatile mathematicians of the twentieth century. Moscow State University.

Can help you study: Probability axioms, Kolmogorov complexity, turbulence, the Kolmogorov-Smirnov test, dynamical systems, and the axiomatic foundations of probability.

Polish-American statistician who, with Egon Pearson, developed the Neyman-Pearson framework for hypothesis testing — the distinction between Type I and Type II errors, the power of a test, and the likelihood ratio. He also invented confidence intervals and the randomised experiment in survey sampling. UC Berkeley, where he built the statistics department.

Can help you study: Hypothesis testing, the Neyman-Pearson lemma, confidence intervals, Type I and Type II errors, statistical power, and the logic of testing scientific claims.

Florence Nightingale David — named after the other Florence Nightingale. English statistician who combined combinatorial mathematics with the history of probability. Her Games, Gods and Gambling (1962) traced probability from dice games in antiquity to Laplace. She worked with Fisher, Neyman, and Pearson, and spent the last two decades of her career at UC Berkeley. She insisted that data without humanity is just numbers.

Can help you study: Combinatorics, the history of probability, experimental statistics, the human dimension of quantitative research, and why the numbers always tell a story about people.

British-American statistician who worked on quality control, time series analysis, experimental design, and Bayesian inference. He coined the phrase that all models are wrong, but some are useful, and he proved it repeatedly in industrial applications. He was Ronald Fisher’s son-in-law. University of Wisconsin-Madison, where he founded the Department of Statistics.

Can help you study: Design of experiments, time series (Box-Jenkins), response surface methodology, quality improvement, Bayesian methods, and the art of building models that are wrong in useful ways.

Pythagoras founded the first mathematical community and proposed that the universe is fundamentally numerical — that reality is made of mathematical relationships rather than material substances. The theorem that bears his name was known before him, but his school established proof as the method of mathematics, as distinct from empirical observation. His discovery (or his school’s discovery) that the square root of two is irrational — that it cannot be expressed as a ratio of whole numbers — reportedly caused a crisis in his community, since it violated the core belief that everything is number. He was also a cult leader who forbade his followers from eating beans.

Can help you with: The foundations of mathematical proof, the Pythagorean theorem and its history, the discovery of irrational numbers, the relationship between mathematics and music (harmonics), early Greek number theory, and the idea that reality is fundamentally mathematical.

Grothendieck transformed twentieth-century mathematics by replacing its objects with its relationships. Where previous algebraic geometry studied geometric objects directly, he replaced them with the functors (structure-preserving maps) between them, discovering that the structure of the relationships contained more information than the objects themselves. His concept of the topos unified geometry, logic, and topology. He resigned from the Institut des Hautes Études Scientifiques in 1970 when he discovered the institute accepted military funding, and eventually withdrew from mathematics altogether, spending his final decades in a remote Pyrenean village writing a 20,000-page spiritual autobiography.

Can help you with: Category theory and its applications, algebraic geometry and schemes, topos theory, the relationship between mathematics and philosophy, the ethics of scientific funding, and what it means to radically transform a field from within before abandoning it.

The greatest mathematician of antiquity: determined bounds for π, computed areas/volumes by a proto-integration method, discovered the law of the lever and the principle of buoyancy. His Method anticipates integral calculus by 1,800 years.

Can help you study: Archimedes’s method of exhaustion, his calculation of π and of areas and volumes, hydrostatics, and his anticipation of calculus.

Reformulated Newtonian mechanics as a purely analytical system in Mécanique analytique, eliminating geometric diagrams and introducing the Lagrangian. Lagrange multipliers and Lagrange’s theorem in group theory bear his name.

Can help you study: Lagrangian mechanics and the calculus of variations, analytical reformulation of Newtonian mechanics, Lagrange multipliers, and contributions to number theory and group theory.

The most broadly productive mathematician of the twentieth century: founded game theory, designed the architecture of modern computers, gave quantum mechanics its mathematical foundations, and contributed to almost every branch of mathematics and its applications.

Can help you study: The von Neumann computer architecture, game theory and minimax, quantum mechanics foundations, operator algebras, and self-replicating automata.

Founded information theory in 1948, defining information as entropy and establishing the channel coding theorem. Cross-posted from Computing.

Can help you study: Information theory, entropy and the bit, channel capacity, the noisy-channel coding theorem, and Boolean circuit design.

Lecturer at Oxford, author as Lewis Carroll of Alice, and a contributor to formal logic, including methods for testing syllogisms with diagrams and minimising logical expressions.

Can help you study: Dodgson’s logical methods, formal reasoning, the testing of syllogisms, and the intersection of mathematics and imagination.

Invented set theory and proved that infinity comes in different sizes, with his diagonal argument showing the real numbers are uncountable. His transfinite numbers are among the most startling creations in mathematics.

Can help you study: Cantor’s diagonal argument, different sizes of infinity, the continuum hypothesis, and the set theory he founded.

The dominant mathematician of his era: axiomatic method, 23 unsolved problems that set the century’s agenda, Hilbert spaces, and the formalist programme that Gödel ended.

Can help you study: Hilbert’s 23 problems, the axiomatic method, Hilbert spaces, foundations of geometry, and the Hilbert programme.

Discovered the paradox that wrecked naive set theory, co-wrote Principia Mathematica to derive mathematics from logic, and spent a lifetime on the philosophical foundations of mathematics.

Can help you study: Russell’s paradox, logicism, Principia Mathematica, type theory, and the philosophical foundations of mathematics.

Proved that any consistent formal system strong enough for arithmetic contains unprovable true statements — and cannot prove its own consistency. The incompleteness theorems ended Hilbert’s programme.

Can help you study: The incompleteness theorems, the limits of formal systems, truth vs provability, Gödel’s completeness theorem, and the consistency of the continuum hypothesis.

Author of the Elements, the most successful mathematical textbook in history. From five postulates it derives hundreds of theorems, establishing the axiomatic method and the model of mathematical proof for two thousand years.

Can help you study: Euclidean geometry, the five postulates, the method of the Elements, and what the discovery of non-Euclidean geometries meant for the status of Euclid.

Named the conic sections, studied their properties comprehensively, and provided the mathematical machinery of epicycles that Ptolemy used to model planetary motion.

Can help you study: The conic sections and Apollonius’s methods, his influence on Kepler, and the epicycle model of the planets.

Discovered the one-sided Möbius strip, introduced barycentric coordinates, and made foundational contributions to projective geometry and to what would become topology.

Can help you study: The Möbius strip and non-orientable surfaces, barycentric coordinates, projective geometry, and the early history of topology.

Gave the lecture that founded differential geometry, introduced the Riemann integral, discovered Riemann surfaces, and formulated the Riemann hypothesis. His geometry is the language of general relativity.

Can help you study: Riemannian geometry, the Riemann hypothesis, Riemann surfaces, the Riemann integral, and Einstein’s use of Riemann’s ideas.

Unified all of geometry under the concept of symmetry groups in the Erlangen Programme, and invented the Klein bottle.

Can help you study: The Erlangen Programme, how different geometries correspond to different symmetry groups, non-Euclidean geometries, and the Klein bottle.

The last mathematical universalist: founded algebraic topology, discovered chaos in the three-body problem, pioneered celestial mechanics.

Can help you study: The Poincaré conjecture, algebraic topology, chaos and the three-body problem, and Poincaré’s philosophy of science.

Discovered the Hopf fibration, proved the Hopf index theorem, and developed homotopy theory. A founding figure of algebraic topology.

Can help you study: The Hopf fibration and fibre bundles, homotopy groups, the index theorem for vector fields, and the early development of homotopy theory.

Dutch artist who, without formal mathematical training, arrived at deep results in symmetry groups, tessellations, hyperbolic geometry (Circle Limit prints), and impossible structures.

Can help you study: The mathematics of Escher’s art: wallpaper groups, hyperbolic geometry, impossible figures, and tessellations.

Invented catastrophe theory (seven elementary forms of discontinuous change), proved foundational theorems of differential topology, and applied topology to biological morphogenesis.

Can help you study: Catastrophe theory and its seven elementary catastrophes, cobordism, transversality, and topology applied to biology.

The first woman mathematician of whose work we have detailed knowledge. Taught in Alexandria, wrote commentaries on Diophantus and Apollonius, edited Ptolemy’s Almagest, and was murdered by a Christian mob in 415 CE. Cross-posted from the Mouseion.

Can help you study: Hypatia’s mathematical and astronomical work, late Neoplatonism, mathematical commentary as scholarship, and her life in Alexandria.