Biographies of the greatest mathematicians are in separate files by birth year:
Born before 400 | Born betw. 400 & 1580 | Born betw. 1580 & 1700 |
Born betw. 1700 & 1800 | Born betw. 1800 & 1850 | Born betw. 1850 & 1940 (this page) |
List of Greatest Mathematicians |
Sofia Vasilyevna Kovalevskaya (1850-1891) Russia
Sofia Kovalevskaya (aka Sonya Kowalevski; née Korvin-Krukovskaya) was initially self-taught, sought out Weierstrass as her teacher, and was later considered the greatest female mathematician ever (before Emma Noether). She was influential in the development of Russian mathematics. Kovalevskaya studied Abelian integrals and partial differential equations, producing the important Cauchy-Kovalevsky Theorem; her application of complex analysis to physics inspired Poincaré and others. Her most famous work was the solution to the Kovalevskaya top, which has been called a "genuine highlight of 19th-century mathematics." Other than the simplest cases solved by Euler and Lagrange, exact ("integrable") solutions to the equations of motion were unknown, so Kovalevskaya received fame and a rich prize when she solved the Kovalevskaya top. Her ingenious solution might be considered a mere curiosity, but since it is still the only post-Lagrange physical motion problem for which an "integrable" solution has been demonstrated, it remains an important textbook example. Kovalevskaya once wrote "It is impossible to be a mathematician without being a poet in soul." She was also a noted playwright.
Jules Henri Poincaré (1854-1912) France
Poincaré founded the theory of algebraic (combinatorial) topology, and is sometimes called the "Father of Topology" (a title also used for Euler and Brouwer). He also did brilliant work in several other areas of mathematics; he was one of the most creative mathematicians ever, and the greatest mathematician of the Constructivist ("intuitionist") style. Poincaré was clumsy and absent-minded; like Galois, he was almost denied admission to French University, passing only because at age 17 he was already far too famous to flunk.In addition to his topology, Poincaré laid the foundations of homology; he discovered automorphic functions (a unifying foundation for the trigonometric and elliptic functions), and essentially founded the theory of periodic orbits; he made major advances in the theory of differential equations. He is credited with partial solution of Hilbert's 22nd Problem. Several important results carry his name, for example the famous Poincaré Recurrence Theorem, which almost seems to contradict the Second Law of Thermodynamics. Poincaré is especially noted for effectively discovering chaos theory, and for posing Poincaré's Conjecture; that conjecture was one of the most famous unsolved problems in mathematics for an entire century, and can be explained without equations to a layman (provided the layman can visualize 3-D surfaces in 4-space). Recently Grigori Perelman proved Poincaré's conjecture, and is eligible for the first Million Dollar math prize in history.
As were most of the greatest mathematicians, Poincaré was intensely interested in physics. He made revolutionary advances in fluid dynamics and celestial motions; he anticipated Minkowski space and much of Einstein's Special Theory of Relativity (including the famous equation E = mc^{2}). Poincaré also found time to become a famous popular writer of philosophy, writing, "Mathematics is the art of giving the same name to different things;" and "A [worthy] mathematician experiences in his work the same impression as an artist; his pleasure is as great and of the same nature;" and "If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living." With his fame, Poincaré helped the world recognize the importance of the new physical theories of Einstein and Planck.
Andrei Andreyevich Markov (1856-1922) Russia
Markov did excellent work in a broad range of mathematics including analysis, number theory, algebra, continued fractions, approximation theory, and especially probability theory: it has been said that his accuracy and clarity transformed probability theory into one of the most perfected areas of mathematics. Markov is best known as the founder of the theory of stochastic processes. In addition to his Ergodic Theorem about such processes, theorems named after him include the Gauss-Markov Theorem of statistics, the Riesz-Markov Theorem of functional analysis, the Markov Brothers' Inequality in the theory of equations, and Markov's Theorem itself, which helps relate the theories of braids and knots to each other. Markov was also noted for his politics, mocking Czarist rule, and insisting that he be excommunicated from the Russian Orthodox Church when Tolstoy was.
Giuseppe Peano (1858-1932) Italy
Giuseppe Peano is one of the most under-appreciated of all great mathematicians. He started his career by proving a fundamental theorem in differential equations, developed practical solution methods for such equations, discovered a continuous space-filling curve (then thought impossible), and laid the foundations of abstract operator theory. He also produced the best calculus textbook of his time, was first to produce a correct (non-paradoxical) definition of surface area, proved an important theorem about Dirichlet functions, did important work in topology, and much more. Much of his work was unappreciated and left for others to rediscover: he anticipated many of Borel's and Lebesgue's results in measure theory, and several concepts and theorems of analysis. He was the champion of counter-examples, and found flaws in published proofs of several important theorems.Most of the preceding work was done when Peano was quite young. Later he focused on mathematical foundations, and this is the work for which he is most famous. He developed rigorous definitions and axioms for set theory, as well as most of the notation of modern set theory. He was first to define arithmetic (and then the rest of mathematics) in terms of set theory. Peano was first to note that some proofs required an explicit Axiom of Choice (although it was Ernst Zermelo who explicitly formulated that Axiom a few years later).
Despite his early show of genius, Peano's quest for utter rigor may have detracted from his influence in mainstream mathematics. Moreover, since he modestly referenced work by predecessors like Dedekind, Peano's huge influence in axiomatic theory is often overlooked. Yet Bertrand Russell reports that it was from Peano that he first learned that a single-member set is not the same as its element; this fact is now taught in elementary school.
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, Hermann Weyl, John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.
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 interment.
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, 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, and calculus of variations.
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. Among the 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 first 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 which showed that atoms' discreteness explained Brownian motion. Another famous 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! 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, he was co-inventor of several devices, including a gyroscopic compass, hearing device, 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.
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. I've chosen to include him on this list because 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.
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 Emma 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." Many would agree that Emmy Noether was the greatest female mathematician ever.
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 solve the four-color problem); 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 of the 20th century. His discovery of gauge invariance and notion of Riemann surfaces form the basis of modern physics. He excelled at many fields including integral equations, harmonic analysis, analytic number theory, Diophantine approximations, and the foundations of mathematics, 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). 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. Because of his contributions to Schrödinger, many think the latter's famous result should be named 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." 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, mathematical physics and other fields. He worked with the Prime Number Theorem and Riemann's Hypothesis, proved that Li(x) underestimates the number of primes infinitely often (although the smallest such example is probably much larger than a googol). 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 replied 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. He 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, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. 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. Later it was noticed that this claim translates to a true statement about 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(). (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan formula; 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.
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, Ramanujan is considered one of the greatest geniuses ever.
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 especially 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; 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; he directly inspired some of 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 first to show a definition of ordered pair as a set. 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, transcendence 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. 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.
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 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'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 a childhood prodigy who could do very complicated mental arithmetic at an early age. As an adult he was noted for hedonism and reckless driving but also became one of the most prolific geniuses 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. He invented cellular automata, famously constructing a self-reproducing automaton. He 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, 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."
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, and a "biased coin" algorithm. 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" 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 Borel-Kolmogorov Paradox, and the intriguing Zero-One Law of "tail events" among random variables.
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."
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.
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 Godel'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. 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 contributions to Bletchley Park's hardware were much less important than his code-breaking algorithms, he did paper designs of two other computers himself, and helped inspire von Neumann's later work. After the war he 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 worked on a broad range of mathematics, most notably algebraic topology and category theory. 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. Often considered one of the two greatest Russian mathematicians ever, 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.
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 Weil conjectures. Selberg is also famous for ground-breaking work on Riemann's Hypothesis, and the first "elementary" proof of the Prime Number Theorem.
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-) 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, 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.
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-) Britain
Atiyah's career has had extraordinary breadth and depth. 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 (Déscartes' 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 has been a key figure in the development of string theory. This work, and Atiyah-inspired work in gauge theory, restored a close relationship between leading edge research in mathematics and physics. Atiyah is known as a vivacious genius in person, inspiring many, e.g. Edward Witten. With Grothendieck retired, Atiyah is often considered to be the greatest living mathematician.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."
John Willard Milnor (1931-) U.S.A.
Milnor has made major advances in topology (especially differential 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.
John Horton Conway (1937-) Britain
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 conjecture, 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 and Archimedes), a nowhere-continuous function that has the Intermediate Value property, the Conway box function, 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.