Biographies of the greatest mathematicians are in separate files by birth year:
Born before 400 | Born betw. 400 & 1559 | Born betw. 1560 & 1699 |
Born betw. 1700 & 1799 | Born betw. 1800 & 1859 | Born betw. 1860 & 1975 (this page) |
List of Greatest Mathematicians |
Samuel Giuseppe Vito Volterra (1860-1946) Italy
Vito Volterra founded the field of functional analysis ('functions of lines'), and used it to extend the work of Hamilton and Jacobi to more areas of mathematical physics. He developed cylindrical waves and the theory of integral equations. He worked in mechanics, developed the theory of crystal dislocations, and was first to propose the use of helium in balloons. Eventually he turned to mathematical biology and made notable contributions to that field, e.g. predator-prey equations.
David Hilbert (1862-1943) Prussia, Germany
Hilbert, often considered the greatest mathematician of the 20th century, was unequaled in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. He proved many new theorems, including the fundamental theorems of algebraic manifolds, and also discovered simpler proofs for older theorems. His examination of calculus led him to the invention of Hilbert space, considered one of the key concepts of functional analysis and modern mathematical physics. His Nullstellensatz Theorem laid the foundation of algebraic geometry. He was a founder of fields like metamathematics and modern logic. He was also the founder of the "Formalist" school which opposed the "Intuitionism" of Kronecker and Brouwer. He developed a new system of definitions and axioms for geometry, replacing the 2200 year-old system of Euclid. As a young Professor he proved his Finite Basis Theorem, now regarded as one of the most important results of general algebra. His mentor, Paul Gordan, had sought the proof for many years, and rejected Hilbert's proof as non-constructive. Later, Hilbert produced the first constructive proof of the Finite Basis Theorem, as well. In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring Theorem.Any one man can only do so much, so the greatest mathematicians should help nurture their colleagues. Hilbert provided a famous List of 23 Unsolved Problems, which inspired and directed the development of 20th-century mathematics. Hilbert was warmly regarded by his colleagues and students, and contributed to the careers of several great mathematicians and physicists including Georg Cantor, Hermann Minkowski, John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein. His doctoral students included Hermann Weyl, Richard Courant, Max Dehn, Teiji Takagi, Ernst Zermelo, Wilhelm Ackermann, the chess champion Emanuel Lasker, and many other famous mathematicians.
Eventually Hilbert turned to physics and made key contributions to classical and quantum physics and to general relativity. He published the Einstein Field Equations independently of Einstein (though his writings make clear he treats this as strictly Einstein's invention).
Hermann Minkowski (1864-1909) Lithuania, Germany
Minkowski won a prestigious prize at age 18 for reconstructing Eisenstein's enumeration of the ways to represent integers as the sum of five squares. (The Paris Academy overlooked that Smith had already published a solution for this!) His proof built on quadratic forms and continued fractions and eventually led him to the new field of Geometric Number Theory, for which Minkowski's Convex Body Theorem (a sort of pigeonhole principle) is often called the Fundamental Theorem. Minkowski was also a major figure in the development of functional analysis. With his "question mark function" and "sausage," he was also a pioneer in the study of fractals. Several other important results are named after him, e.g. the Hasse-Minkowski Theorem. He was first to extend the Separating Axis Theorem to multiple dimensions. Minkowski was one of Einstein's teachers, and also a close friend of David Hilbert. He is particularly famous for building on Poincaré's work to invent Minkowski space to deal with Einstein's Special Theory of Relativity. This not only provided a better explanation for the Special Theory, but helped inspire Einstein toward his General Theory. Minkowski said that his "views of space and time ... have sprung from the soil of experimental physics, and therein lies their strength.... Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
Jacques Salomon Hadamard (1865-1963) France
Hadamard made revolutionary advances in several different areas of mathematics, especially complex analysis, analytic number theory, differential geometry, partial differential equations, symbolic dynamics, chaos theory, matrix theory, and Markov chains; for this reason he is sometimes called the "Last Universal Mathematician." He also made contributions to physics. One of the most famous results in mathematics is the Prime Number Theorem, that there are approximately n/log n primes less than n. This result was conjectured by Legendre and Gauss, attacked cleverly by Riemann and Chebyshev, and finally, by building on Riemann's work, proved by Hadamard and Vallee-Poussin. (Hadamard's proof is considered more elegant and useful than Vallee-Poussin's.) Several other important theorems are named after Hadamard (e.g. his Inequality of Determinants), and some of his theorems are named after others (Hadamard was first to prove Brouwer's Fixed-Point Theorem for arbitrarily many dimensions). Hadamard was also influential in promoting others' work: He is noted for his survey of Poincaré's work; his staunch defense of the Axiom of Choice led to the acceptance of Zermelo's work. Hadamard was a successful teacher, with André Weil, Maurice Fréchet, and others acknowledging him as key inspiration. Like many great mathematicians he emphasized the importance of intuition, writing "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it."
Felix Hausdorff (1868-1942) Germany
Hausdorff had diverse interests: he composed music and wrote poetry, studied astronomy, wrote on philosophy, but eventually focused on mathematics, where he did important work in several fields including set theory, measure theory, functional analysis, and both algebraic and point-set topology. His studies in set theory led him to the Hausdorff Maximal Principle, and the Generalized Continuum Hypothesis; his concepts now called Hausdorff measure and Hausdorff dimension led to geometric measure theory and fractal geometry; his Hausdorff paradox led directly to the famous Banach-Tarski paradox; he introduced other seminal concepts, e.g. Hausdorff Distance. He also worked in analysis, solving the Hausdorff moment problem.As Jews in Hitler's Germany, Hausdorff and his wife committed suicide rather than submit to internment.
Élie Joseph Cartan (1869-1951) France
Cartan worked in the theory of Lie groups and Lie algebras, applying methods of topology, geometry and invariant theory to Lie theory, and classifying all Lie groups. This work was so significant that Cartan, rather than Lie, is considered the most important developer of the theory of Lie groups. Using Lie theory and ideas like his Method of Prolongation he advanced the theories of differential equations and differential geometry. Cartan introduced several new concepts including algebraic group, exterior differential forms, spinors, moving frames, Cartan connections. He proved several important theorems, e.g. Schläfli's Conjecture about embedding Riemann metrics, Stokes' Theorem, and fundamental theorems about symmetric Riemann spaces. He made a key contribution to Einstein's general relativity, based on what is now called Riemann-Cartan geometry. Cartan's methods were so original as to be fully appreciated only recently; many now consider him to be one of the greatest mathematicians of his era. In 1938 Weyl called him "the greatest living master in differential geometry."
Félix Édouard Justin Émile Borel (1871-1956) France
Borel exhibited great talent while still in his teens, soon practically founded modern measure theory, and received several honors and prizes. Among his famous theorems is the Heine-Borel Covering Theorem. He also did important work in several other fields of mathematics, including divergent series, quasi-analytic functions, differential equations, number theory, complex analysis, theory of functions, geometry, probability theory, and game theory. Relating measure theory to probabilities, he introduced concepts like normal numbers and the Borel-Kolmogorov paradox. He also did work in relativity and the philosophy of science. He anticipated the concept of chaos, inspiring Poincaré. Borel combined great creativity with strong analytic power; however he was especially interested in applications, philosophy, and education, so didn't pursue the tedium of rigorous development and proof; for this reason his great importance as a theorist is often underestimated. Borel was decorated for valor in World War I, entered politics between the Wars, and joined the French Resistance during World War II.
Tullio Levi-Civita (1873-1941) Italy
Levi-Civita was noted for strong geometrical intuition, and excelled at both pure mathematics and mathematical physics. He worked in analytic number theory, differential equations, tensor calculus, hydrodynamics, celestial mechanics, and the theory of stability. Several inventions are named after him, e.g. the non-archimedean Levi-Civita field, the Levi-Civita parallelogramoid, and the Levi-Civita symbol. His work inspired all three of the greatest 20th-century mathematical physicists, laying key mathematical groundwork for Weyl's unified field theory, Einstein's relativity, and Dirac's quantum theory.
Henri Léon Lebesgue (1875-1941) France
Lebesgue did groundbreaking work in real analysis, advancing Borel's measure theory; his Lebesgue integral superseded the Riemann integral and improved the theoretical basis for Fourier analysis. Several important theorems are named after him, e.g. the Lebesgue Differentiation Theorem and Lebesgue's Number Lemma. He did important work on Hilbert's 19th Problem, and in the Jordan Curve Theorem for higher dimensions. In 1916, the Lebesgue integral was compared "with a modern Krupp gun, so easily does it penetrate barriers which were impregnable." In addition to his seminal contributions to measure theory and Fourier analysis, Lebesgue made significant contributions in several other fields including complex analysis, topology, set theory, potential theory, dimension theory, and calculus of variations.
Edmund Georg Hermann Landau (1877-1938) Germany
Landau was one of the most prolific and influential number theorists ever and wrote the first comprehensive treatment of analytic number theory. He was also adept at complex function theory. He was especially keen at finding very simple proofs: one of his most famous results was a simpler proof of Hadamard's prime number theorem; being simpler it was also more fruitful and led to Landau's Prime Ideal Theorem. In addition to simpler proofs of existing theorems, new theorems by Landau include important facts about Riemann's Hypothesis; facts about Dirichlet series; key lemmas of analysis; a result in Waring's Problem; a generalization of the Little Picard Theorem; and a partial proof of Gauss' conjecture about the density of classes of composite numbers. He developed key results in probabilistic number theory (e.g. the Landau-Ramanujan constant) before Hardy and Ramanujan did. In 1912 Landau described four conjectures about prime numbers which were 'unattackable with present knowledge': (a) Goldbach's conjecture, (b) infinitely many primes n^2+1, (c) infinitely many twin primes (p, p+2), (d) a prime exists in every interval (n^2, n^2+n). By 2018 none of these conjectures have been resolved, though much progress has been made in each case. Landau was the inventor of big-O notation. Hardy wrote that no one was ever more passionately devoted to mathematics than Landau.
Godfrey Harold Hardy (1877-1947) England
Hardy was an extremely prolific research mathematician who did important work in analysis (especially the theory of integration), number theory, global analysis, and analytic number theory. He proved several important theorems about numbers, for example that Riemann's zeta function has infinitely many zeros with real part 1/2. He was also an excellent teacher and wrote several excellent textbooks, as well as a famous treatise on the mathematical mind. He abhorred applied mathematics, treating mathematics as a creative art; yet his work has found application in population genetics, cryptography, thermodynamics and particle physics.Hardy is especially famous (and important) for his encouragement of and collaboration with Ramanujan. Hardy provided rigorous proofs for several of Ramanujan's conjectures, including Ramanujan's "Master Theorem" of analysis. Among other results of this collaboration was the Hardy-Ramanujan Formula for partition enumeration, which Hardy later used as a model to develop the Hardy-Littlewood Circle Method; Hardy then used this method to prove stronger versions of the Hilbert-Waring Theorem, and in prime number theory; the method has continued to be a very productive tool in analytic number theory. Hardy was also a mentor to Norbert Wiener, another famous prodigy.
Hardy once wrote "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." He also wrote "Beauty is the first test; there is no permanent place in the world for ugly mathematics."
René Maurice Fréchet (1878-1973) France
Maurice Fréchet introduced the concept of metric spaces (though not using that term); and also made major contributions to point-set topology. Building on work of Hadamard and Volterra, he generalized Banach spaces to use new (non-normed) metrics and proved that many important theorems still applied in these more general spaces. For this work, and his invention of the notion of compactness, Fréchet is called the Founder of the Theory of Abstract Spaces. He also did important work in probability theory and in analysis; for example he proved the Riesz Representation Theorem the same year Riesz did. Many theorems and inventions are named after him, for example Fréchet Distance, which has many applications in applied math, e.g. protein structure analysis.
Albert Einstein (1879-1955) Germany, Switzerland, U.S.A.
Albert Einstein was unquestionably one of the two greatest physicists in all of history. The atomic theory achieved general acceptance only after Einstein's 1905 paper, introducing the Einstein-Smoluchowski relation, showed that atoms' discreteness explained Brownian motion. Another 1905 paper introduced the famous equation E = mc^{2}; yet Einstein published other papers that same year, two of which were more important and influential than either of the two just mentioned. No wonder that physicists speak of the Miracle Year without bothering to qualify it as Einstein's Miracle Year! (Before his Miracle Year, Einstein had been a mediocre undergraduate, and held minor jobs including patent examiner.) Altogether Einstein published at least 300 books or papers on physics. For example, in a 1917 paper he anticipated the principle of the laser. Also, sometimes in collaboration with Leo Szilard, he was co-inventor of several devices, including a gyroscopic compass, hearing aid, automatic camera and, most famously, the Einstein-Szilard refrigerator. He became a very famous and influential public figure. (For example, it was his letter that led Roosevelt to start the Manhattan Project.) Among his many famous quotations is: "The search for truth is more precious than its possession."Einstein is most famous for his Special and General Theories of Relativity, but he should be considered the key pioneer of Quantum Theory as well, drawing inferences from Planck's work that no one else dared to draw. Indeed it was his articulation of the quantum principle in a 1905 paper which has been called "the most revolutionary sentence written by a physicist of the twentieth century." Einstein's discovery of the photon in that paper led to his only Nobel Prize; years later, he was first to call attention to the "spooky" nature of quantum entanglement. Einstein was also first to call attention to a flaw in Weyl's earliest unified field theory. But despite the importance of his other contributions it is Einstein's General Theory which is his most profound contribution. Minkowski had developed a flat 4-dimensional space-time to cope with Einstein's Special Theory; but it was Einstein who had the vision to add curvature to that space to describe acceleration.
Einstein certainly has the breadth, depth, and historical importance to qualify for this list; but his genius and significance were not in the field of pure mathematics. (He acknowledged his limitation, writing "I admire the elegance of your [Levi-Civita's] method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.") Einstein was a mathematician, however; he pioneered the application of tensor calculus to physics and invented the Einstein summation notation. That Einstein's Equation of General Relativity explained a discrepancy in Mercury's orbit was a discovery made by Einstein personally (a discovery he described as 'joyous excitement' that gave him heart palpitations). He composed a beautiful essay about mathematical proofs using the Theorem of Menelaus as his example. The sheer strength and diversity of his intellect is suggested by his elegant paper on river meanders, a classic of that field. Certainly he belongs on a Top 100 List: his extreme greatness overrides his focus away from math. Einstein ranks #10 on Michael Hart's famous list of the Most Influential Persons in History. His General Theory of Relativity has been called the most creative and original scientific theory ever. Einstein once wrote "... the creative principle resides in mathematics [; thus] I hold it true that pure thought can grasp reality, as the ancients dreamed."
Oswald Veblen (1880-1960) U.S.A.
Oswald Veblen's first mathematical achievement was a novel system of axioms for geometry. He also worked in topology; projective geometry; differential geometry (where he was first to introduce the concept of differentiable manifold); ordinal theory (where he introduced the Veblen hierarchy); and mathematical physics where he worked with spinors and relativity. He developed a new theory of ballistics during World War I and helped plan the first American computer during World War II. His famous theorems include the Veblen-Young Theorem (an important algebraic fact about projective spaces); a proof of the Jordan Curve Theorem more rigorous than Jordan's; and Veblen's Theorem itself (a generalization of Euler's result about cycles in graphs). Veblen, a nephew of the famous economist Thorstein Veblen, was an important teacher; his famous students included Alonzo Church, John W. Alexander, Robert L. Moore, and J.H.C. Whitehead. He was also a key figure in establishing Princeton's Institute of Advanced Study; the first five mathematicians he hired for the Institute were Einstein, von Neumann, Weyl, J.W. Alexander and Marston Morse.
Luitzen Egbertus Jan Brouwer (1881-1966) Holland
Brouwer is often considered the "Father of Topology;" among his important theorems were the Fixed Point Theorem, the "Hairy Ball" Theorem, the Jordan-Brouwer Separation Theorem, and the Invariance of Dimension. He developed the method of simplicial approximations, important to algebraic topology; he also did work in geometry, set theory, measure theory, complex analysis and the foundations of mathematics. He was first to anticipate forms like the Lakes of Wada, leading eventually to other measure-theory "paradoxes." Several great mathematicians, including Weyl, were inspired by Brouwer's work in topology.Brouwer is most famous as the founder of Intuitionism, a philosophy of mathematics in sharp contrast to Hilbert's Formalism, but Brouwer's philosophy also involved ethics and aesthetics and has been compared with those of Schopenhauer and Nietzsche. Part of his mathematics thesis was rejected as "... interwoven with some kind of pessimism and mystical attitude to life which is not mathematics ..." As a young man, Brouwer spent a few years to develop topology, but once his great talent was demonstrated and he was offered prestigious professorships, he devoted himself to Intuitionism, and acquired a reputation as eccentric and self-righteous.
Intuitionism has had a significant influence, although few strict adherents. Since only constructive proofs are permitted, strict adherence would slow mathematical work. This didn't worry Brouwer who once wrote: "The construction itself is an art, its application to the world an evil parasite."
Amalie Emmy Noether (1882-1935) Germany
Noether was an innovative researcher who was considered the greatest master of abstract algebra ever; her advances included a new theory of ideals, the inverse Galois problem, and the general theory of commutative rings. She originated novel reasoning methods, especially one based on "chain conditions," which advanced invariant theory and abstract algebra; her insistence on generalization led to a unified theory of modules and Noetherian rings. Her approaches tended to unify disparate areas (algebra, geometry, topology, logic) and led eventually to modern category theory. Her invention of Betti homology groups led to algebraic topology, and thus revolutionized topology.Noether's work has found various applications in physics, and she made direct advances in mathematical physics herself. Noether's Theorem establishing that certain symmetries imply conservation laws has been called the most important Theorem in physics since the Pythagorean Theorem. Several other important theorems are named after her, e.g. Noether's Normalization Lemma, which provided an important new proof of Hilbert's Nullstellensatz. Noether was an unusual and inspiring teacher; her successful students included Emil Artin, Max Deuring, Jacob Levitzki, etc. She was generous with students and colleagues, even allowing them to claim her work as their own. Noether was close friends with the other greatest mathematicians of her generation: Hilbert, von Neumann, and Weyl. Weyl once said he was embarrassed to accept the famous Professorship at Göttingen because Noether was his "superior as a mathematician." Emmy Noether is considered the greatest female mathematician ever.
Waclaw Sierpinski (1882-1969) Poland
Sierpinski won a gold medal as an undergraduate by making a major improvement to a famous theorem by Gauss about lattice points inside a circle. He went on to do important research in set theory, number theory, point set topology, the theory of functions, and fractals. He was extremely prolific, producing 50 books and over 700 papers. He was a Polish patriot: he contributed to the development of Polish mathematics despite that his land was controlled by Russians or Nazis for most of his life. He worked as a code-breaker during the Polish-Soviet War, helping to break Soviet ciphers. Conditionally convergent series are intriguing; Sierpinski proved a stronger form of the Riemann rearrangement theorem, that such series can be rearranged to converge to any chosen real value.Sierpinski was first to prove Tarski's remarkable conjecture that the Generalized Continuum Hypothesis implies the Axiom of Choice. He developed three famous fractals: a space-filling curve; the Sierpinski gasket; and the Sierpinski carpet, which covers the plane but has area zero and has found application in antennae design. Borel had proved that almost all real numbers are "normal" but Sierpinski was the first to actually display a number which is normal in every base. He proved the existence of infinitely many Sierpinski numbers having the property that, e.g. (78557·2^{n}+1) is a composite number for every natural number n. It remains an unsolved problem (likely to be defeated soon with high-speed computers) whether 78557 is the smallest such "Sierpinski number."
Solomon Lefschetz (1884-1972) Russia, U.S.A.
Lefschetz was born in Russia, educated as an engineer in France, moved to U.S.A., was severely handicapped in an accident, and then switched to pure mathematics. He was a key founder of algebraic topology, even coining the word topology, and pioneered the application of topology to algebraic geometry. Starting from Poincaré's work, he developed Lefschetz duality and used it to derive conclusions about fixed points in topological mappings. The Lefschetz Fixed-point Theorem left Brouwer's famous result as just a special case. His Picard-Lefschetz theory eventually led to the proof of the Weil conjectures. Lefschetz also did important work in algebraic geometry, non-linear differential equations, and control theory. As a teacher he was noted for a combative style. Preferring intuition over rigor, he once told a student who had improved on one of Lefschetz's proofs: "Don't come to me with your pretty proofs. We don't bother with that baby stuff around here."
George David Birkhoff (1884-1984) U.S.A.
Birkhoff is one of the greatest native-born American mathematicians ever, and did important work in many fields. There are several significant theorems named after him: the Birkhoff-Grothendieck Theorem is an important result about vector bundles; Birkhoff's Theorem is an important result in algebra; and Birkhoff's Ergodic Theorem is a key result in statistical mechanics which has since been applied to many other fields. His Poincaré-Birkhoff Fixed Point Theorem is especially important in celestial mechanics, and led to instant worldwide fame: the great Poincaré had described it as most important, but had been unable to complete the proof. In algebraic graph theory, he invented Birkhoff's chromatic polynomial (while trying to prove the four-color map theorem); he proved a significant result in general relativity which implied the existence of black holes; he also worked in differential equations and number theory; he authored an important text on dynamical systems. Like several of the great mathematicians of that era, Birkhoff developed his own set of axioms for geometry; it is his axioms that are often found in today's high school texts. Birkhoff's intellectual interests went beyond mathematics; he once wrote "The transcendent importance of love and goodwill in all human relations is shown by their mighty beneficent effect upon the individual and society."
Hermann Klaus Hugo (Peter) Weyl (1885-1955) Germany, U.S.A.
Weyl studied under Hilbert and became one of the premier mathematicians and thinkers of the 20th century. Along with Hilbert and Poincaré he was a great "universal" mathematician; his discovery of gauge invariance and notion of Riemann surfaces form the basis of modern physics; he was also a creative thinker in philosophy. Weyl excelled at many fields of mathematics including integral equations, harmonic analysis, analytic number theory, Diophantine approximations, axiomatic theory, and mathematical philosophy; but he is most respected for his revolutionary advances in geometric function theory (e.g., differentiable manifolds), the theory of compact groups (incl. representation theory), and theoretical physics (e.g., Weyl tensor, gauge field theory and invariance). His theorems include key lemmas and foundational results in several fields; Atiyah commented that whenever he explored a new topic he found that Weyl had preceded him. Although he was a master of algebra, he revealed his philosophic preference by writing "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." For a while, Weyl was a disciple of Brouwer's Intuitionism and helped advance that doctrine, but he eventually found it too restrictive. Weyl was also a very influential figure in all three major fields of 20th-century physics: relativity, unified field theory and quantum mechanics. He and Einstein were great admirers of each other. Because of his contributions to Schrödinger, many think the latter's famous result should be named the Schrödinger-Weyl Wave Equation.
Vladimir Vizgin wrote "To this day, Weyl's [unified field] theory astounds all in the depth of its ideas, its mathematical simplicity, and the elegance of its realization." The Nobel prize-winner Julian Schwinger, himself considered an inscrutable genius, was so impressed by Weyl's book connecting quantum physics to group theory that he likened Weyl to a "god" because "the ways of gods are mysterious, inscrutable, and beyond the comprehension of ordinary mortals." Weyl once wrote: "My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful."
John Edensor Littlewood (1885-1977) England
John Littlewood was a very prolific researcher. (This fact is obscured somewhat in that many papers were co-authored with Hardy, and their names were always given in alphabetic order.) The tremendous span of his career is suggested by the fact that he won Smith's Prize (and Senior Wrangler) in 1905 and the Copley Medal in 1958. He specialized in analysis and analytic number theory but also did important work in combinatorics, Fourier theory, Diophantine approximations, differential equations, and other fields. He also did important work in practical engineering, creating a method for accurate artillery fire during the First World War, and developing equations for radio and radar in preparation for the Second War. He worked with the Prime Number Theorem and Riemann's Hypothesis; and proved the unexpected fact that Chebyshev's bias, and Li(x)>π(x), while true for most, and all but very large, numbers, are violated infinitely often. (Building on this result, it is now known that there is a big patch of primes near 10^{9608} that exceed the Li(x) prediction, though few if any of those primes are actually known.) Some of his work was elementary, e.g. his elegant proof that a cube cannot be dissected into unequal cubes; but most of his results were too specialized to state here, e.g. his widely-applied 4/3 Inequality which guarantees that certain bimeasures are finite, and which inspired one of Grothendieck's most famous results. Hardy once said that his friend was "the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power." Littlewood's response was that it was possible to be too strong of a mathematician, "forcing through, where another might be driven to a different, and possibly more fruitful, approach."
Srinivasa Ramanujan Iyengar (1887-1920) India
Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. While some of these were old theorems or just curiosities, many were brilliant new theorems with very difficult proofs. For example, he found a beautiful identity connecting Poisson summation to the Möbius function. He also found a brilliant generalization of Lagrange's Four Square Theorem; and much more. Ramanujan might be almost unknown today, except that his letter caught the eye of Godfrey Hardy, who saw remarkable, almost inexplicable formulae which "must be true, because if they were not true, no one would have had the imagination to invent them." Ramanujan's specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, the divisor function, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. Ramanujan's "Master Theorem" has wide application in analysis, and has been applied to the evaluation of Feynman diagrams. Much of his best work was done in collaboration with Hardy, for example a proof that almost all numbers n have about log log n prime factors (a result which developed into probabilistic number theory). Much of his methodology, including unusual ideas about divergent series, was his own invention. As a young man he made the absurd claim that 1+2+3+4+... = -1/12; his method for this is now called Ramanujan summation. Later it was noticed that this particular Ramanujan sum corresponds with the summation Euler had developed to regularize the Riemann zeta function (with which Ramanujan was unfamiliar). Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi.Ramanujan's most famous work was with the partition enumeration function p(), Hardy guessing that some of these discoveries would have been delayed at least a century without Ramanujan. Together, Hardy and Ramanujan developed an analytic approximation to p(), although Hardy was initially awed by Ramanujan's intuitive certainty about the existence of such a formula, and even the form it would have. (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan approximation; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.)
In a letter from his deathbed, Ramanujan introduced his mysterious "mock theta functions", gave examples, and developed their properties. Much later these forms began to appear in disparate areas: combinatorics, the proof of Fermat's Last Theorem, and even knot theory and the theory of black holes. It was only recently, more than 80 years after Ramanujan's letter, that his conjectures about these functions were proven; solutions mathematicians had sought unsuccessfully were found among his examples. Mathematicians are baffled that Ramanujan could make these conjectures, which they confirmed only with difficulty using methods not available in Ramanujan's day.
Many of Ramanujan's results are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.) Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. (Ramanujan may have had unrecorded proofs, poverty leading him to use chalk and erasable slate rather than paper.) Unlike Abel, much of whose work depended on the complex numbers, most of Ramanujan's work focused on real numbers. Despite these limitations, some consider Ramanujan to be the greatest mathematical genius ever; but he ranks as low as #17 since many lesser mathematicians were much more influential.
Because of its fast convergence, an odd-looking formula of Ramanujan is sometimes used to calculate π:
99^{2} / π = √8 ∑_{k=0,∞} ((4k)! (1103+26390 k) / (k!^{4} 396^{4k}))
Thoralf Albert Skolem (1887-1963) Norway
Thoralf Skolem proved fundamental theorems of lattice theory, proved the Skolem-Noether Theorem of algebra, also worked with set theory and Diophantine equations; but is best known for his work in logic, metalogic, and non-standard models. Some of his work preceded similar results by Gödel. He developed a theory of recursive functions which anticipated some computer science. He worked on the famous Löwenheim-Skolem Theorem which has the "paradoxical" consequence that systems with uncountable sets can have countable models. ("Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he considered an absurdity, nondenumerable sets being, for him, fictions without real existence.")
George Pólya (1887-1985) Hungary
George Pólya (Pólya György) did significant work in several fields: complex analysis, probability, geometry, algebraic number theory, and combinatorics, but is most noted for his teaching How to Solve It, the craft of problem posing and proof. He is also famous for the Pólya Enumeration Theorem. Several other important theorems he proved include the Pólya-Vinogradov Inequality of number theory, the Pólya-Szego Inequality of functional analysis, and the Pólya Inequality of measure theory. He introduced the Hilbert-Pólya Conjecture that the Riemann Hypothesis might be a consequence of spectral theory (in 2017 this Conjecture was partially proved by a team of physicists, and the Riemann Hypothesis might be close to solution!). He introduced the famous "All horses are the same color" example of inductive fallacy; he named the Central Limit Theorem of statistics. Pólya was the "teacher par excellence": he wrote top books on multiple subjects; his successful students included John von Neumann. His work on plane symmetry groups directly inspired Escher's drawings. Having huge breadth and influence, Pólya has been called "the most influential mathematician of the 20th century."
Stefan Banach (1892-1945) Poland
Stefan Banach was a self-taught mathematician who is most noted as the "Founder of Functional Analysis" and for his contributions to measure theory. Among several important theorems bearing his name are the Uniform Boundedness (Banach-Steinhaus) Theorem, the Open Mapping (Banach-Schauder) Theorem, the Contraction Mapping (Banach fixed-point) Theorem, and the Hahn-Banach Theorem. Many of these theorems are of practical value to modern physics; however he also proved the paradoxical Banach-Tarski Theorem, which demonstrates a sphere being rearranged into two spheres of the same original size. (Banach's proof uses the Axiom of Choice and is sometimes cited as evidence that that Axiom is false.) The wide range of Banach's work is indicated by the Banach-Mazur results in game theory (which also challenge the axiom of choice). Banach also made brilliant contributions to probability theory, set theory, analysis and topology.Banach once said "Mathematics is the most beautiful and most powerful creation of the human spirit."
Norbert Wiener (1894-1964) U.S.A.
Norbert Wiener entered college at age 11, studying various sciences; he wrote a PhD dissertation at age 17 in philosophy of mathematics where he was one of the first to show a definition of ordered pair as a set. (Hausdorff also proposed such a definition; both Wiener's and Hausdorff's definitions have been superseded by Kuratowski's (a, b) = {{a}, {a, b}} despite that it leads to a singleton when a=b.) He then did important work in several topics in applied mathematics, including stochastic processes (beginning with Brownian motion), potential theory, Fourier analysis, the Wiener-Hopf decomposition useful for solving differential and integral equations, communication theory, cognitive science, and quantum theory. Many theorems and concepts are named after him, e.g the Wiener Filter used to reduce the error in noisy signals. His most important contribution to pure mathematics was his generalization of Fourier theory into generalized harmonic analysis, but he is most famous for his writings on feedback in control systems, for which he coined the new word, cybernetics. Wiener was first to relate information to thermodynamic entropy, and anticipated the theory of information attributed to Claude Shannon. He also designed an early analog computer. Although they differed dramatically in both personal and mathematical outlooks, he and John von Neumann were the two key pioneers (after Turing) in computer science. Wiener applied his cybernetics to draw conclusions about human society which, unfortunately, remain largely unheeded.
Carl Ludwig Siegel (1896-1981) Germany
Carl Siegel became famous when his doctoral dissertation established a key result in Diophantine approximations. He continued with contributions to several branches of analytic and algebraic number theory, including arithmetic geometry and quadratic forms. He also did seminal work with Riemann's zeta function, Dedekind's zeta functions, transcendental number theory, discontinuous groups, the 3-body problem in celestial mechanics, and symplectic geometry. In complex analysis he developed Siegel modular forms, which have wide application in math and physics. He may share credit with Alexander Gelfond for the solution to Hilbert's 7th Problem. Siegel admired the "simplicity and honesty" of masters like Gauss, Lagrange and Hardy and lamented the modern "trend for senseless abstraction." He and Israel Gelfand were the first two winners of the Wolf Prize in Mathematics. Atle Selberg called him a "devastatingly impressive" mathematician who did things that "seemed impossible." André Weil declared that Siegel was the greatest mathematician of the first half of the 20th century.
Pavel Sergeevich Aleksandrov (1896-1962) Russia
Aleksandrov worked in set theory, metric spaces and several fields of topology, where he developed techniques of very broad application. He pioneered the studies of compact and bicompact spaces, and homology theory. He laid the groundwork for a key theorem of metrisation. His most famous theorem may be his discovery about "perfect subsets" when he was just 19 years old. Much of his work was done in collaboration with Pavel Uryson and Heinz Hopf. Aleksandrov was an important teacher; his students included Lev Pontryagin.
Emil Artin (1898-1962) Austria, Germany, U.S.A.
Artin was an important and prolific researcher in several fields of algebra, including algebraic number theory, the theory of rings, field theory, algebraic topology, Galois theory, a new method of L-series, and geometric algebra. Among his most famous theorems were Artin's Reciprocity Law, key lemmas in Galois theory, and results in his Theory of Braids. He also produced two very influential conjectures: his conjecture about the zeta function in finite fields developed into the field of arithmetic geometry; Artin's Conjecture on primitive roots inspired much work in number theory, and was later generalized to become Weil's Conjectures. He is credited with solution to Hilbert's 17th Problem and partial solution to the 9th Problem. His prize-winning students include John Tate and Serge Lang. Artin also did work in physical sciences, and was an accomplished musician.
Paul Adrien Maurice Dirac (1902-1984) England, U.S.A.
Dirac had a severe father and was bizarrely taciturn (perhaps autistic), but became one of the greatest mathematical physicists ever. He developed Fermi-Dirac statistics, applied quantum theory to field theory, predicted the existence of magnetic monopoles, and was first to note that some quantum equations lead to inexplicable infinities. His most important contribution was to combine relativity and quantum mechanics by developing, with pure thought, the Dirac Equation. From this equation, Dirac deduced the existence of anti-electrons, a prediction considered so bizarre it was ignored -- until anti-electrons were discovered in a cloud chamber four years later. For this work he was awarded the Nobel Prize in Physics at age 31, making him one of the youngest Laureates ever. Dirac's mathematical formulations, including his Equation and the Dirac-von Neumann axioms, underpin all of modern particle physics. After his great discovery, Dirac continued to do important work, some of which underlies modern string theory. He was also adept at more practical physics; although he declined an invitation to work on the Manhattan Project, he did contribute a fundamental result in centrifuge theory to that Project.The Dirac Equation was one of the most important scientific discoveries of the 20th century and Dirac was certainly a superb mathematical genius, but I've left Dirac off of the Top 100 since his work didn't advance "pure" mathematics. Like many of the other greatest mathematical physicists (Kepler, Einstein, Weyl), Dirac thought the true equations of physics must have beauty, writing "... it is more important to have beauty in one's equations than to have them fit experiment ... [any discrepancy may] get cleared up with further development of the theory."
Alfred Tarski (1902-1983) Poland, U.S.A.
Alfred Tarski (born Alfred Tajtelbaum) was one of the greatest and most prolific logicians ever, but also made advances in set theory, measure theory, topology, algebra, group theory, computability theory, metamathematics, and geometry. He was also acclaimed as a teacher. Although he achieved fame at an early age with the Banach-Tarski Paradox, his greatest achievements were in formal logic. He wrote on the definition of truth, developed model theory, and investigated the completeness questions which also intrigued Gödel. He proved several important systems to be incomplete, but also established completeness results for real arithmetic and geometry. His most famous result may be Tarski's Undefinability Theorem, which is related to Gödel's Incompleteness Theorem but more powerful. Several other theorems, theories and paradoxes are named after Tarski including Tarski-Grothendieck Set Theory, Tarski's Fixed-Point Theorem of lattice theory (from which the famous Cantor-Bernstein-Schröder Theorem is a simple corollary), and a new derivation of the Axiom of Choice (which Lebesgue refused to publish because "an implication between two false propositions is of no interest"). Tarski was first to enunciate the remarkable fact that the Generalized Continuum Hypothesis implies the Axiom of Choice, although proof had to wait for Sierpinski. Tarski's other notable accomplishments include his cylindrical algebra, ordinal algebra, universal algebra, and an elegant and novel axiomatic basis of geometry.
John von Neumann (1903-1957) Hungary, U.S.A.
John von Neumann (born Neumann Janos Lajos) was an amazing childhood prodigy who could do very complicated mental arithmetic and much more at an early age. One of his teachers burst into tears at their first meeting, astonished that such a genius existed. As an adult he was noted for hedonism and reckless driving but also became one of the most prolific thinkers in history, making major contributions in many branches of both pure and applied mathematics. He was an essential pioneer of both quantum physics and computer science.Von Neumann pioneered the use of models in set theory, thus improving the axiomatic basis of mathematics. He proved a generalized spectral theorem sometimes called the most important result in operator theory. He developed von Neumann Algebras. He was first to state and prove the Minimax Theorem and thus invented game theory; this work also advanced operations research; and led von Neumann to propose the Doctrine of Mutual Assured Destruction which was a basis for Cold War strategy. He developed cellular automata (first invented by Stanislaw Ulam), famously constructing a self-reproducing automaton. He worked in mathematical foundations: he formulated the Axiom of Regularity and invented elegant definitions for the counting numbers (0 = {}, n+1 = n ∪ {n}). He also worked in analysis, matrix theory, measure theory, numerical analysis, ergodic theory (discovering Birkhoff's Ergodic Theorem before Birkhoff did), group representations, continuous geometry, statistics and topology. Von Neumann discovered an ingenious area-conservation paradox related to the famous Banach-Tarski volume-conservation paradox. He inspired some of Gödel's famous work (and independently proved Gödel's Second Theorem). He is credited with (partial) solution to Hilbert's 5th Problem using the Haar Theorem; this also relates to quantum physics. George Pólya once said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper." Michael Atiyah has said he calls only three people geniuses: Wolfgang Mozart, Srinivasa Ramanujan, and Johnny von Neumann.
Von Neumann did very important work in fields other than pure mathematics. By treating the universe as a very high-dimensional phase space, he constructed an elegant mathematical basis (now called von Neumann algebras) for the principles of quantum physics. He advanced philosophical questions about time and logic in modern physics. He played key roles in the design of conventional, nuclear and thermonuclear bombs; he also advanced the theory of hydrodynamics. He applied game theory and Brouwer's Fixed-Point Theorem to economics, becoming a major figure in that field. His contributions to computer science are many: in addition to co-inventing the stored-program computer, he was first to use pseudo-random number generation, finite element analysis, the merge-sort algorithm, a "biased coin" algorithm, and (though Ulam first conceived the approach) Monte Carlo simulation. By implementing wide-number software he joined several other great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus, Madhava, and (by proxy), Ramanujan) in producing the best approximation to π of his time. At the time of his death, von Neumann was working on a theory of the human brain.
Andrey Nikolaevich Kolmogorov (1903-1987) Russia
Kolmogorov had a powerful intellect and excelled in many fields. As a youth he dazzled his teachers by constructing toys that appeared to be "Perpetual Motion Machines." At the age of 19, he achieved fame by finding a Fourier series that diverges almost everywhere, and decided to devote himself to mathematics. He is considered the founder of the fields of intuitionistic logic, algorithmic complexity theory, and (by applying measure theory) modern probability theory. He also excelled in topology, set theory, trigonometric series, and random processes. He and his student Vladimir Arnold proved the surprising Superposition Theorem, which not only solved Hilbert's 13th Problem, but went far beyond it. He and Arnold also developed the "magnificent" Kolmogorov-Arnold-Moser (KAM) Theorem, which quantifies how strong a perturbation must be to upset a quasiperiodic dynamical system. Kolmogorov's axioms of probability are considered a partial solution of Hilbert's 6th Problem. He made important contributions to the constructivist ideas of Kronecker and Brouwer. While Kolmogorov's work in probability theory had direct applications to physics, Kolmogorov also did work in physics directly, especially the study of turbulence. There are dozens of notions named after Kolmogorov, such as the Kolmogorov Backward Equation, the Chapman-Kolmogorov equations, the Borel-Kolmogorov Paradox, and the intriguing Zero-One Law of "tail events" among random variables.
Henri Paul Cartan (1904-2008) France
Henri Cartan, son of the great Élie Cartan, is particularly noted for his work in algebraic topology, and analytic functions; but also worked with sheaves, and many other areas of mathematics. He was a key member of the Bourbaki circle. (That circle was led by Weil, emphasized rigor, produced important texts, and introduced terms like in-, sur-, and bi-jection, as well as the Ø symbol.) Working with Samuel Eilenberg (also a Bourbakian), Cartan advanced the theory of homological algebra. He is most noted for his many contributions to the theory of functions of several complex variables. Henri Cartan was an important influence on Grothendieck and others, and an excellent teacher; his students included Jean-Pierre Serre.
Kurt Gödel (1906-1978) Germany, U.S.A.
Gödel, who had the nickname Herr Warum ("Mr. Why") as a child, was perhaps the foremost logic theorist ever, clarifying the relationships between various modes of logic. He partially resolved both Hilbert's 1st and 2nd Problems, the latter with a proof so remarkable that it was connected to the drawings of Escher and music of Bach in the title of a famous book. He was a close friend of Albert Einstein, and was first to discover "paradoxical" solutions (e.g. time travel) to Einstein's equations. About his friend, Einstein later said that he had remained at Princeton's Institute for Advanced Study merely "to have the privilege of walking home with Gödel." (Like a few of the other greatest 20th-century mathematicians, Gödel was very eccentric.)Two of the major questions confronting mathematics are: (1) are its axioms consistent (its theorems all being true statements)?, and (2) are its axioms complete (its true statements all being theorems)? Gödel turned his attention to these fundamental questions. He proved that first-order logic was indeed complete, but that the more powerful axiom systems needed for arithmetic (constructible set theory) were necessarily incomplete. He also proved that the Axioms of Choice (AC) and the Generalized Continuum Hypothesis (GCH) were consistent with set theory, but that set theory's own consistency could not be proven. He may have established that the truths of AC and GCH were independent of the usual set theory axioms, but the proof was left to Paul Cohen.
In Gödel's famous proof of Incompleteness, he exhibits a true statement (G) which cannot be proven, to wit "G (this statement itself) cannot be proven." If G could be proven it would be a contradictory true statement, so consistency dictates that it indeed cannot be proven. But that's what G says, so G is true! This sounds like mere word play, but building from ordinary logic and arithmetic Gödel was able to construct statement G rigorously.
André Weil (1906-1998) France, U.S.A.
Weil made profound contributions to several areas of mathematics, especially algebraic geometry, which he showed to have deep connections with number theory. His Weil conjectures were very influential; these and other works laid the groundwork for some of Grothendieck's work. Weil proved a special case of the Riemann Hypothesis; he contributed, at least indirectly, to the recent proof of Fermat's Last Theorem; he also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representation theory, and theta functions. He invented several new concepts including vector bundles, and uniform space. His work has found applications in particle physics and string theory. He is considered to be one of the most influential of modern mathematicians.Weil's biography is interesting. He studied Sanskrit as a child, loved to travel, taught at a Muslim university in India for two years (intending to teach French civilization), wrote as a young man under the famous pseudonym Nicolas Bourbaki, spent time in prison during World War II as a Jewish objector, was almost executed as a spy, escaped to America, and eventually joined Princeton's Institute for Advanced Studies. He once wrote: "Every mathematician worthy of the name has experienced [a] lucid exaltation in which one thought succeeds another as if miraculously."
Shiing-Shen Chern (1911-2004) China, U.S.A.
Shiing-Shen Chern (Chen Xingshen) studied under Élie Cartan, and became perhaps the greatest master of differential geometry. He is especially noted for his work in algebraic geometry, topology and fiber bundles, developing his Chern characters (in a paper with "a tremendous number of geometrical jewels"), developing Chern-Weil theory, the Chern-Simons invariants, and especially for his brilliant generalization of the Gauss-Bonnet Theorem to multiple dimensions. His work had a major influence in several fields of modern mathematics as well as gauge theories of physics. Chern was an important influence in China and a highly renowned and successful teacher: one of his students (Yau) won the Fields Medal, another (Yang) the Nobel Prize in physics. Chern himself was the first Asian to win the prestigious Wolf Prize.
Alan Mathison Turing (1912-1954) Britain
Turing developed a new foundation for mathematics based on computation; he invented the abstract Turing machine, designed a "universal" version of such a machine, proved the famous Halting Theorem (related to Gödel's Incompleteness Theorem), and developed the concept of machine intelligence (including his famous Turing Test proposal). He also introduced the notions of definable number and oracle (important in modern computer science), and was an early pioneer in the study of neural networks. For this work he is called the Father of Computer Science and Artificial Intelligence. Turing also worked in group theory, numerical analysis, and complex analysis; he developed an important theorem about Riemann's zeta function; he had novel insights in quantum physics. During World War II he turned his talents to cryptology; his creative algorithms were considered possibly "indispensable" to the decryption of German Naval Enigma coding, which in turn is judged to have certainly shortened the War by at least two years. Although his clever code-breaking algorithms were his most spectacular contributions at Bletchley Park, he was also a key designer of the Bletchley "Bombe" computer. After the war he helped design other physical computers, as well as theoretical designs; and helped inspire von Neumann's later work. He (and earlier, von Neumann) wrote about the Quantum Zeno Effect which is sometimes called the Turing Paradox. He also studied the mathematics of biology, especially the Turing Patterns of morphogenesis which anticipated the discovery of BZ reactions. Turing's life ended tragically: charged with immorality and forced to undergo chemical castration, he apparently took his own life. With his outstanding depth and breadth, Alan Turing would qualify for our list in any event, but his decisive contribution to the war against Hitler gives him unusually strong historic importance.
Paul Erdös (1913-1996) Hungary, U.S.A., Israel, etc.
Erdös was a childhood prodigy who became a famous (and famously eccentric) mathematician. He is best known for work in combinatorics (especially Ramsey Theory) and partition calculus, but made contributions across a very broad range of mathematics, including graph theory, analytic number theory, probabilistic methods, and approximation theory. He is regarded as the second most prolific mathematician in history, behind only Euler. Although he is widely regarded as an important and influential mathematician, Erdös founded no new field of mathematics: He was a "problem solver" rather than a "theory developer." He's left us several still-unproven intriguing conjectures, e.g. that 4/n = 1/x + 1/y + 1/z has positive-integer solutions for any n.Erdös liked to speak of "God's Book of Proofs" and discovered new, more elegant, proofs of several existing theorems, including the two most famous and important about prime numbers: Chebyshev's Theorem that there is always a prime between any n and 2n, and (though the major contributor was Atle Selberg) Hadamard's Prime Number Theorem itself. He also proved many new theorems, such as the Erdös-Szekeres Theorem about monotone subsequences with its elegant (if trivial) pigeonhole-principle proof.
Samuel Eilenberg (1913-1998) Poland, U.S.A.
Eilenberg is considered a founder of category theory, but also worked in algebraic topology, automata theory and other areas. He coined several new terms including functor, category, and natural isomorphism. Several other concepts are named after him, e.g. a proof method called the Eilenberg telescope or Eilenberg-Mazur Swindle. He worked on cohomology theory, homological algebra, etc. By using his category theory and axioms of homology, he unified and revolutionized topology. Most of his work was done in collaboration with others, e.g. Henri Cartan; but he also single-authored an important text laying a mathematical foundation for theories of computation and language. Sammy Eilenberg was also a noted art collector.
Israel Moiseevich Gelfand (1913-2009) Russia
Gelfand was a brilliant and important mathematician of outstanding breadth with a huge number of theorems and discoveries. He was a key figure of functional analysis and integral geometry; he pioneered representation theory, important to modern physics; he also worked in many fields of analysis, soliton theory, distribution theory, index theory, Banach algebra, cohomology, etc. He made advances in physics and biology as well as mathematics. He won the Order of Lenin three times and several prizes from Western countries. Considered one of the two greatest Russian mathematicians of the 20th century, the two were compared with "[arriving in a mountainous country] Kolmogorov would immediately try to climb the highest mountain; Gelfand would immediately start to build roads." In old age Israel Gelfand emigrated to the U.S.A. as a professor, and won a MacArthur Fellowship.
Claude Elwood Shannon (1916-2001) U.S.A.
Shannon's initial fame was for a paper called "possibly the most important master's thesis of the century." That paper founded digital circuit design theory by proving that universal computation was achieved with an ensemble of switches and boolean gates. He also worked with analog computers, theoretical genetics, and sampling and communication theories. Early in his career Shannon was fortunate to work with several other great geniuses including Weyl, Turing, Gödel and even Einstein; this may have stimulated him toward a broad range of interests and expertise. He was an important and prolific inventor, discovering signal-flow graphs, the topological gain formula, etc.; but also inventing the first wearable computer (to time roulette wheels in Las Vegas casinos), a chess-playing algorithm, a flame-throwing trumpet, and whimsical robots (e.g. a "mouse" that navigated a maze). His hobbies included juggling, unicycling, blackjack card-counting. His investigations into gambling theory led to new approaches to the stock market.Shannon worked in cryptography during World War II; he was first to note that a one-time pad allowed unbreakable encryption as long as the pad was as large as the message; he is also noted for Shannon's maxim that a code designer should assume the enemy knows the system. His insights into cryptology eventually led to information theory, or the mathematical theory of communication, in which Shannon established the relationships among bits, entropy, power and noise. It is as the Founder of Information Theory that Shannon has become immortal.
Atle Selberg (1917-2007) Norway, U.S.A.
Selberg may be the greatest analytic number theorist ever. He also did important work in Fourier spectral theory, lattice theory (e.g. introducing and partially proving the conjecture that "all lattices are arithmetic"), and the theory of automorphic forms, where he introduced Selberg's Trace Formula. He developed a very important result in analysis called the Selberg Integral. Other Selberg techniques of general utility include mollification, sieve theory, and the Rankin-Selberg method. These have inspired other mathematicians, e.g. contributing to Deligne's proof of the Weil conjectures. Selberg is also famous for ground-breaking work on Riemann's Hypothesis, and the first "elementary" proof of the Prime Number Theorem.
John Torrence Tate (1925-) U.S.A.
Tate, a student of Emil Artin, is a master of algebraic number theory, p-adic theory and arithmetic geometry. Using Fourier analysis and Tate cohomology groups, he revolutionized the treatments of class field theory and algebraic K-theory. In addition to Tate cohomology groups, Tate's key inventions include rigid analytic geometry, Hodge-Tate theory, Tate-Barsotti groups, applications of adele ring self-duality, the Tate module, Tate curve, Tate twists, and much more. His long and productive career earned the Abel Prize for his "vast and lasting impact on the theory of numbers [and] his incisive contributions and illuminating insights ... He has truly left a conspicuous imprint on modern mathematics."
Jean-Pierre Serre (1926-) France
Serre did important work with spectral sequences and algebraic methods, revolutionizing the study of algebraic topology and algebraic geometry, especially homotopy groups and sheaves. Hermann Weyl praised Serre's work strongly, saying it gave an important new algebraic basis to analysis. He collaborated with Grothendieck and Pierre Deligne, helped resolve the Weil conjectures, and contributed indirectly to the recent proof of Fermat's Last Theorem. His wide range of research areas also includes number theory, bundles, fibrations, p-adic modular forms, Galois representation theory, and more. Serre has been much honored: he is the youngest ever to win a Fields Medal; 49 years after his Fields Medal he became the first recipient of the Abel Prize.
Alexandre Grothendieck (1928-2014) Germany, France
Grothendieck has done brilliant work in several areas of mathematics including number theory, geometry, topology, and functional analysis, but especially in the fields of algebraic geometry and category theory, both of which he revolutionized. He is especially noted for his invention of the Theory of Schemes, and other methods to unify different branches of mathematics. He applied algebraic geometry to number theory; applied methods of topology to set theory; etc. Grothendieck is considered a master of abstraction, rigor and presentation. He has produced many important and deep results in homological algebra, most notably his etale cohomology. With these new methods, Grothendieck and his outstanding student Pierre Deligne were able to prove the Weil Conjectures. Grothendieck also developed the theory of sheafs, the theory of motives, generalized the Riemann-Roch Theorem to revolutionize K-theory, developed Grothendieck categories, crystalline cohomology, infinity-stacks and more. The guiding principle behind much of Grothendieck's work has been Topos Theory, which he invented to harness the methods of topology. These methods and results have redirected several diverse branches of modern mathematics including number theory, algebraic topology, and representation theory. Among Grothendieck's famous results was his Fundamental Theorem in the Metric Theory of Tensor Products, which was inspired by Littlewood's proof of the 4/3 Inequality.Grothendieck's radical religious and political philosophies led him to retire from public life while still in his prime, but he is widely regarded as the greatest mathematician of the 20th century, and indeed one of the greatest geniuses ever.
John Forbes Nash, Jr. (1928-2015) U.S.A
The Riemann Embedding Problems were important puzzles of geometry that baffled many of the greatest minds for a century. Hilbert showed that Lobachevsky's hyperbolic plane could not be embedded into Euclidean 3-space, but what about into Euclidean 4-space? Cartan and Chern were among the great mathematicians who solved various special cases, but using "methods entirely without precedent" John Nash demonstrated a general solution. This was a true highlight of 20th-century mathematics.Nash was a lonely, tormented schizophrenic whose life was portrayed in the film Beautiful Mind. He achieved early fame with his work in game theory; this eventually led to the Nobel Prize in Economics. Earlier studies in game theory focused on the simplest cases (two-person zero-sum, or cooperative), but Nash demonstrated "Nash equilibria" for n-person or non-zero-sum non-cooperative games. Nash also excelled at several other fields of mathematics, especially topology, algebraic geometry, partial differential equations, elliptic functions, and the theory of manifolds (including singularity theory, the concept of real algebraic manifolds and isotropic embeddings). He proved theorems of great importance which had defeated all earlier attempts. His most famous theorems were the Nash Embedding Theorems, e.g. that any Riemannian manifold of dimension k can be embedded isometrically into some n-dimensional Euclidean space. Other important work was in partial differential equations where he proved that strong regularity constraints apply to solutions of the equations of heat and fluid flow.
Lennart Axel Edvard Carleson (1928-) Sweden
Carleson is a master of complex analysis, especially harmonic analysis, and dynamical systems; he proved many difficult and important theorems; among these are a theorem about quasiconformal mapping extension, a technique to construct higher dimensional strange attractors, and the famous Kakutani Corona Conjecture, whose proof brought Carleson great fame. For the Corona proof he introduced Carleson measures, one of several useful tools he's created for his masterful proofs. In 1966, four years after proving Kakutani's Conjecture, he proved the 53-year old Luzin's Conjecture, a strong statement about Fourier convergence. This was startling because of a 38-year old conjecture suggested by Kolmogorov that Luzin's Conjecture was false.
Michael Francis (Sir) Atiyah (1929-2019) Britain
Atiyah's career had extraordinary breadth and depth; he was sometimes called the greatest English mathematician since Isaac Newton. He advanced the theory of vector bundles; this developed into topological K-theory and the Atiyah-Singer Index Theorem. This Index Theorem is considered one of the most far-reaching theorems ever, subsuming famous old results (Descartes' total angular defect, Euler's topological characteristic), important 19th-century theorems (Gauss-Bonnet, Riemann-Roch), and incorporating important work by Weil and especially Shiing-Shen Chern. It is a key to the study of high-dimension spaces, differential geometry, and equation solving. Several other key results are named after Atiyah, e.g. the Atiyah-Bott Fixed-Point Theorem, the Atiyah-Segal Completion Theorem, and the Atiyah-Hirzebruch spectral sequence. Atiyah's work developed important connections not only between topology and analysis, but with modern physics; Atiyah himself was a key figure in the development of string theory; and was a proponent of the recent idea that octonions may underlie particle physics. He also studied the physics of instantons and monopoles. This work, and Atiyah-inspired work in gauge theory, restored a close relationship between leading edge research in mathematics and physics. His interest in physics, and an old theory of von Neumann, led him, as a very old man, to explore the fine structure constant of physics. In September 2018 he announced that this recent work would lead to a proof of the Riemann Hypothesis but other mathematicians are quite skeptical. Nonetheless, Michael Atiyah is still regarded as one of the very greatest mathematicians of the 20th century.Atiyah was known as a vivacious genius in person, inspiring many, e.g. Edward Witten. Atiyah once said a mathematician must sometimes "freely float in the atmosphere like a poet and imagine the whole universe of possibilities, and hope that eventually you come down to Earth somewhere else." He also said "Beauty is an important criterion in mathematics ... It determines what you regard as important and what is not."
John Willard Milnor (1931-) U.S.A.
Milnor founded the field of differential topology and has made other major advances in topology, algebraic geometry and dynamical systems. He discovered Milnor maps (related to fiber bundles); important theorems in knot theory; the Duality Theorem for Reidemeister Torsion; the Milnor Attractors of dynamical systems; a new elegant proof of Brouwer's "Hairy Ball" Theorem; and much more. He is especially famous for two counterexamples which each revolutionized topology. His "exotic" 7-dimensional hyperspheres gave the first examples of homeomorphic manifolds that were not also diffeomorphic, and developed the fields of differential topology and surgery theory. Milnor invented certain high-dimensional polyhedra to disprove the Hauptvermutung ("main conjecture") of geometric topology. While most famous for his exotic counterexamples, his revolutionary insights into dynamical systems have important value to practical applied mathematics. Although Milnor has been called the "Wizard of Higher Dimensions," his work in dynamics began with novel insights into very low-dimensional systems.As Fields, Presidential and (twice) Putnam Medalist, as well as winner of the Abel, Wolf and two Steele Prizes; Milnor can be considered the most "decorated" mathematician of the modern era.
Roger Penrose is a thinker of great breadth, who has contributed to biology and philosophy, as well as to mathematics, general relativity and cosmology. Some of his earliest work was done in collaboration with his father Lionel, a polymath and professor of psychiatry who developed the Penrose Square Root Law of voting theory. Together, Roger and his father discovered the 'impossible tri-bar' and an impossible staircase which inspired work by the artist M.C. Escher. And, in turn, Escher's drawings may have helped inspire Penrose's most famous discoveries in recreational mathematics: non-periodic tilings. He soon found such a tiling with just two tile shapes; the previous record was six shapes. (Nine years after that, such tilings were observed in nature as "quasi-crystals.") Penrose has written several successful popular books on science.As a mathematician, Penrose did important work in algebra: he was first to conceive of the generalized matrix inverse, and used it for novel solutions in linear algebra and spectral decomposition. He did more important work in geometry and topology; for example, he proved theorems about embedding (or "unknotting") manifolds in Euclidean space. His best mathematics, e.g. the invention of twistor theory, was inspired by his pursuit of Einstein's general relativity.
Penrose is most noted for his very creative work in cosmology, specifically in the mathematics of gravitation, space-time, black holes and the Big Bang. He developed new methods to apply spinors and Riemann tensors to gravitation. His twistor theory was an effort to relate general relativity to quantum theory; this work advanced both physics and mathematics. The top physicist Kip Thorne said "Roger Penrose revolutionized the mathematical tools that we use to analyse the properties of space-time." Stephen Hawking was an early convert to Penrose's methods; the mathematical laws of black holes (and the Big Bang) are called the Penrose-Hawking Singularity Theorems. Penrose formulated the Censorship Hypotheses about black holes, e.g. the Riemannian Penrose Inequality and the Weyl Curvature Hypothesis; and discovered Penrose-Terrell rotation.
Penrose has proposed conformal cyclic cosmology, that in the entropy death of one universe, the scaling of time and distance become arbitrary and the dying universe becomes the big bang for another. Recently it is proposed that evidence for this can be seen in the details of the cosmic microwave background radiation from the early universe. Many of his theories are extremely controversial: He claims that Gödel's Incompleteness Theorem provides insight into human consciousness. He has developed a detailed theory that quantum effects (involving the microtubules in neurons) enhance the capability of biologic brains.
John Griggs Thompson (1932-) U.S.A.
Thompson is the master of finite groups. He achieved early fame by proving a long-standing conjecture about Frobenius groups. He followed up by proving (with Walter Feit) that all nonabelian finite simple groups are of even order. This result, proved in a 250-page paper, stunned the world of mathematics; it led to the classification of all finite groups. Thompson also made major contributions to coding theory, and to the inverse Galois problem. His work with Galois groups has been called "the most important advance since Hilbert's time."
Robert Phelan Langlands (1936-) Canada, U.S.A.
Langlands started by studying semigroups and partial differential equations but soon switched his attention to representation theory where he found deep connections between group theory and automorphic forms; he then used these connections to make profound discoveries in number theory. Langlands' methods, collectively called the Langlands Program, are now central to all of these fields. The Langlands Dual Group ^{L}G revolutionized representation theory and led to a large number of conjectures. One of these conjectures is the Principle of Functoriality, of which a partial proof allowed Langlands to prove a famous conjecture of Artin, and Wiles to prove Fermat's Last Theorem. Langlands and others have applied these methods to prove several other old conjectures, and to formulate new more powerful conjectures. He has also worked with Eisenstein series, L-functions, Lie groups, percolation theory, etc. He mentored several important mathematicians (including Thomas Hales, mentioned briefly in Pappus' mini-bio).Langlands once wrote "Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world." He was appointed Hermann Weyl Professor at the Institute for Advanced Study and now sits in the office once occupied by Albert Einstein. This seems appropriate since, as the man "who reinvented mathematics," his advances have sometimes been compared to Einstein's.
Vladimir Igorevich Arnold (1937-2010) Russia
Arnold is most famous for solving Hilbert's 13th Problem; for the "magnificent" Kolmogorov-Arnold-Moser (KAM) Theorem; and for "Arnold diffusion," which identifies exceptions to the stability promised by the KAM Theorem. In addition to dynamical systems theory, Arnold made contributions to catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, mathematical physics; and was the essential founder of modern singularity theory.
John Horton Conway (1937-) Britain, U.S.A.
Conway has done pioneering work in a very broad range of mathematics including knot theory, number theory, group theory, lattice theory, combinatorial game theory, geometry, quaternions, tilings, and cellular automaton theory. He started his career by proving a case of Waring's Problem, but achieved fame when he discovered the largest then-known sporadic group (the symmetry group of the Leech lattice); this sporadic group is now known to be second in size only to the Monster Group, with which Conway also worked. Conway's fertile creativity has produced a cornucopia of fascinating inventions: markable straight-edge construction of the regular heptagon (a feat also achieved by Alhazen, Thabit, Vieta and perhaps Archimedes), a nowhere-continuous function that has the Intermediate Value property, the Conway box function, the rational tangle theorem in knot theory, the aperiodic pinwheel tiling, a representation of symmetric polyhedra, the silly but elegant Fractran programming language, his chained-arrow notation for large numbers, and many results and conjectures in recreational mathematics. He found the simplest proof for Morley's Trisector Theorem (sometimes called the best result in simple plane geometry since ancient Greece). He proved an unusual theorem about quantum physics: "If experimenters have free will, then so do elementary particles." His most famous construction is the computationally complete automaton known as the Game of Life. His most important theoretical invention, however, may be his surreal numbers incorporating infinitesimals; he invented them to solve combinatorial games like Go, but they have pure mathematical significance as the largest possible ordered field.Conway's great creativity and breadth certainly make him one of the greatest living mathematicians. Conway has won the Nemmers Prize in Mathematics, and was first winner of the Pólya Prize.
Mikhael Leonidovich Gromov (1943-) Russia, France
Gromov is considered one of the greatest geometers ever, but he has a unique "soft" approach to geometry which leads to applications in other fields: Gromov has contributed to group theory, partial differential equations, other areas of analysis and algebra, and even mathematical biology. He is especially famous for his pseudoholomorphic curves; they revolutionized the study of symplectic manifolds and are important in string theory. By applying his geometric ideas to all areas of mathematics, Gromov has become one of the most influential living mathematicians. He has proved a very wide variety of theorems: important results about groups of polynomial growth, theorems essential to Perelman's proof of the Poincaré conjecture, the nonsqueezing theorem of Hamiltonian mechanics, theorems of systolic geometry, and various inequalities and compactness theorems. Several concepts are named after him, including Gromov-Hausdorff convergence, Gromov-Witten invariants, Gromov's random groups, Gromov product, etc.
Pierre René Deligne (1944-) Belgium, France, U.S.A.
Using new ideas about cohomology, in 1974 Pierre Deligne stunned the world of mathematics with a spectacular proof of the Weil conjectures. Proof of these conjectures, which were key to further progress in algebraic geometry, had eluded the great Alexandre Grothendieck. With his "unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity," Deligne made other important contributions to a broad range of mathematics in addition to algebraic geometry, including algebraic and analytic number theory, topology, group theory, the Langlands conjectures, Grothendieck's theory of motives, and Hodge theory. Deligne also found a partial solution of Hilbert's 21st Problem. Several ideas are named after him including Deligne-Lusztig theory, Deligne-Mumford stacks, Fourier-Deligne transform, the Langlands-Deligne local constant, Deligne cohomology, and at least eight distinct conjectures.
Shelah has advanced logic, model theory, set theory, and especially the theory of cardinal numbers. His work has led to new methods in diverse fields like group theory, topology, measure theory, stability theory, algebraic geometry, Banach spaces, and combinatorics. He solved outstanding problems like Morley's Problem; proved the independence of the Whitehead problem; found a "Jonsson group" and counterexamples to outstanding conjectures; improved on Arrow's Voting Theorem; and, most famously, proved key results about singular cardinals.Shelah is the founder of the theories of proper forcing, classification of models, and possible cofinalities. He has authored over 800 papers and several books, making him one of the most prolific mathematicians in history. He has been described as a "phenomenal mathematician, ... produc[ing] results at a furious pace."
William Paul Thurston (1946-2012) U.S.A.
Thurston revolutionized the study of 3-D manifolds; it was his work which eventually led Perelman to a proof of Poincaré's Conjecture. He developed the key results of foliation theory and, with Gromov, co-founded geometric group theory. One of his award citations states "Thurston has fantastic geometric insight and vision: his ideas have completely revolutionized the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry."
Witten is the world's greatest living physicist, and one of the top mathematicians. Not only does Witten apply mathematics to solve problems in physics, but his broad knowledge of physics, especially quantum field theory, string theory, supersymmetry, and quantum gravity, has led him to novel connections and insights in abstract geometry and topology, as well as physics. His skill with string theory led him to a novel theory of invariants and allowed him to improve results in knot theory; his skill with supersymmetry led him to new results in differential geometry. He has applied quantum field theory to higher-dimensional spaces and found new insights there. He has proven several important new theorems of mathematics and general relativity but also has had unproven insights which inspired proofs by others. His discovery that the five competing models of string theory were all congruent to a single 'M-theory' (sometimes called a 'Theory of Everything') revolutionized string theory. Several theorems or concepts are named after Witten, including Seiberg-Witten theory, the Weinberg-Witten theorem, the Gromov-Witten invariant, the Witten index, Witten conjecture, Witten-type Topological quantum field theory, etc.Witten started his college career studying fields like history and linguistics. When he finally switched to math and physics he learned at breathtaking speed. His fellows do not compare him to other living mathematical physicists; they compare him to Einstein, Weyl, Newton and Ramanujan.
Terence Chi-Shen Tao (1975-) Australia, U.S.A.
Tao was a phenomenal child prodigy who has become one of the most admired living mathematicians. He has made important contributions to partial differential equations, combinatorics, harmonic analysis, number theory, group theory, model theory, nonstandard analysis, random matrices, the geometry of 3-manifolds, functional analysis, ergodic theory, etc. and areas of applied math including quantum mechanics, general relativity, and image processing. He has been called the first since David Hilbert to be expert across the entire spectrum of mathematics. Among his earliest important discoveries were results about the multi-dimensional Kakeya needle problem, which led to advances in Fourier analysis and fractals. In addition to his numerous research papers he has written many highly regarded textbooks. One of his prize citations commends his "sheer technical power, his other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view."Paul Erdös mentored Tao when he was a ten-year old prodigy, and the two are frequently compared. They are both prolific problem solvers across many fields, though have founded no new fields. As with Erdös, much of Tao's work has been done in collaboration: for example with Van Vu he proved the circular law of random matrices; with Ben Green he proved the Dirac-Motzkin conjecture and solved the "orchard-planting problem." Especially famous is the Green-Tao Theorem that there are arbitrarily long arithmetic series among the prime numbers (or indeed among any sufficiently dense subset of the primes). This confirmed an old conjecture by Lagrange, and was especially remarkable because the proof fused methods from number theory, ergodic theory, harmonic analysis, discrete geometry, and combinatorics. Tao is also involved in recent efforts to attack the famous Twin Prime Conjecture.