Biographies

NOTE: In this section we have sometimes gone into more detail about interesting people who are not so well known, and been satisfied with brief mention of very famous mathematicians. (This is not an exhoustive list of everyone mentioned in the book).

Much of this information was obtained by searching the World Wide Web (Internet). Thanks are due to Google and to the indispensable web sites Wikipedia, MacTutor (written and edited by J. J. O’Connor and E. F. Robertson at the University of St. Andrews), and Mathematicians of the African Diaspora (maintained by Scott W. Williams).

Ralph Abraham (1936–). U.S. mathematician actively involved in the development of the theory of dynamical systems in the 1960s and 1970s.

Jean d’Alembert (1717–1783). French mathematician who was a pioneer in the study of differential equations and their use in physics. He studied the equilibrium and motion of fluids.

P. S. Aleksandrov (1896–1982). Russian topologist who wrote about 300 scientific works in his long career. He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929.

Richard D. Anderson (1922–2008). U.S. mathematician who was a student of Robert Lee Moore. His work at first centered around the geometric topology of continua. He subsequently was largely responsible, along with his students, for developing infinite-dimensional topology.

Archimedes of Syracuse (c. 287 BC–c. 212 BC). Outstanding genius physicist and mathematician of the classical era.

Vladimir I. Arnold (1937–2010). Russian mathematician who while still a teen-aged student of Andrei Kolmogorov at Moscow State University, solved Hilbert’s 13th problem, by showing that any continuous function of several variables can be constructed with a finite number of two-variable functions. Since then he has made major contributions to an astounding number of different mathematical disciplines.

Michael Atiyah (1929–). Lebanese-British mathematician widely considered one of the greatest geometers of the 20th century. In the 1960s his path-breaking work with Isadore Singer produced the Atiyah-Singer index theorem, a result that helped to develop several branches of mathematics. Earlier, together with Friedrich Hirzebruch, he founded the study of another major tool in algebraic topology: topological K-theory. It was inspired by Alexander Grothendieck’s work on the Riemann-Roch theorem and has since generated algebraic K-theory.

John Carlos Baez (1961–). U.S. mathematical physicist at the University of California, Riverside. He is known for his work on loop quantum gravity and on applications of higher categories to physics. His sister Joan is a famous singer.

Stefan Banach (1892–1945). Polish mathematician who founded modern functional analysis and made major contributions on topological vector spaces, measure theory, integration, and orthogonal series.

Henri Baruk (1897–1999). French neuropsychiatrist. Baruk spent his childhood living among patients at the asylum where his father Jacques was the director.

Edwin Beckenbach (1906–1982). U.S. mathematician who contributed to the creation of the Institute for Numerical Analysis at UCLA in 1948. Its SWAC computing machine was one of the half-dozen most powerful computers in the world. The mathematicians who gathered to use it made UCLA known around the world.

Edward Griffith Begle (1914–1978). U.S. mathematician who was the director of the School Mathematics Study Group (SMSG), the group mainly credited for developing the “new math.”

Eric Temple Bell (1883–1960). Scottish-American number theorist and prolific author. His Men of Mathematics is a very widely read collection of mathematical biographies.

Felix Bernstein (1878–1956). German statistician and mathematician who taught at Göttingen from 1907 to 1934. In 1921 he founded the Institute of Mathematical Statistics, and in 1934 he emigrated to the United States. He returned to Göttingen in 1948. He published a famous theorem on the equivalence of sets while at Cantor’s seminar at Halle in 1897.

Lipman Bers (1914–1993). Latvian-American mathematician who worked on Riemann surfaces. He received his Ph.D. in 1938 from the University of Prague under Charles Loewner. He was a much loved and admired mentor of graduate students and an outstanding defender of human rights internationally.

Abram Samoilovitch Besicovitch (1891–1970). Russian-Jewish mathematician who studied under A. A. Markov at St. Petersburg University. He converted to Eastern Orthodoxy, joining the Russian Orthodox Church, on marrying in 1916. In 1924 he joined Harald Bohr in Copenhagen, where he worked on almost periodic functions, which now bear his name. He moved to Cambridge in 1927, where he was appointed to the Rouse Ball Chair of Mathematics, which he held until his retirement in 1958. He worked mainly on combinatorial methods and questions in real analysis, such as the Kakeya needle problem and the Hausdorff-Besicovitch dimension.

Enrico Betti (1823–1892). Italian mathematician who taught at the University of Pisa and was noted for contributions to algebra and topology. Betti also did important work in theoretical physics, in particular on potential theory and elasticity.

R H Bing (1914–1986). U.S. mathematician, student of Robert Lee Moore, who worked on general topology, particularly on metrization, and on planar sets, where he examined webs, cuttings, and planar embeddings.

George David Birkhoff (1884–1944). First leading U.S. mathematician educated in the United States, at Chicago and Harvard. His most important work was the ergodic theorem he proved in 1931.

Joan S. Lyttle Birman (1927–). U.S. mathematician. After years as a systems analyst in the aircraft industry, she took a break to raise three children. In 1961 she began working with Wilhelm Magnus and in 1968 received a Ph.D. from the Courant Institute of Mathematical Sciences. Birman’s mathematical work has focused on low-dimensional topology: braids, knots, surface mappings, and 3-manifolds.

David Blackwell (1919–2010). Professor emeritus of statistics at the University of California, Berkeley, and one of the eponyms of the Rao-Blackwell theorem. In 1965 he was the first African American to be inducted into the National Academy of Sciences. David Blackwell said that the work that gave him the most satisfaction was infinite games and analytic sets. He found a game theory proof of the Kuratowski reduction theorem connecting the areas of game theory and topology.

André Bloch (1893–1948). French mathematician who is remembered for a result about univalent functions called Bloch’s theorem. All his mathematical output was produced while he was confined in an institution for the criminally insane.

Lenore Blum (1942–). U.S. mathematician and logician teaching at Carnegie-Mellon. She described her work as follows: “Continuity is the mathematics of calculus and physics but there’s never been a theory of computation that deals with this continuum.”

Ralph Philip Boas, Jr. (1912–1992). U.S. mathematician, teacher, and journal editor at Northwestern University who wrote over 200 papers mainly in the fields of real and complex analysis.

Harald Bohr (1887–1951). Danish mathematician who worked on Dirichlet series and applied analysis to the theory of numbers. He is the only mathematician to win an Olympic medal (on Denmark’s soccer team in 1908). He was the brother of the great physicist Niels Bohr.

Béla Bollobás (1943–). Hungarian-British graph theorist at Cambridge University. After earning a doctorate in discrete geometry in 1967, he spent a year in Moscow with I. M. Gelfand, and then in 1972 received a second doctorate, in functional analysis, from Cambridge.

János Bolyai (1802–1860). Hungarian mathematician who was one of the three famous discoverers of non-Euclidean geometry along with Gauss and Lobatchevsky.

Nicolas Bourbaki. Pseudonym of a group of (mainly) French mathematicians, established in 1935, who dominated much of pure mathematics in the 1950s and 1960s.

L. E. J. Brouwer (1881–1966). Dutch mathematician best known for his topological fixed point theorem. He founded the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are governed by self-evident laws.

Felix E. Browder (1927–). U.S. mathematician known for his work on elliptic partial differential equations. He is the brother of the mathematicians William and Andrew Browder. He received a doctorate from Princeton University in 1948. Browder was the recipient of the 1999 National Medal of Science. He also served as president of the American Mathematical Society from 1999 to 2000.

Justine Bumby. U.S. mathematician who lived with Alexandre Grothendieck. Their son is the statistician John Grothendieck.

Georg Cantor (1845–1918). German mathematician who founded set theory, which is considered by some to be the foundation of mathematics. He introduced the concept of infinite cardinal and ordinal numbers.

Lazare Nicolas Marguérite Carnot (1753–1823). French mathematician best known as a geometer. In 1803 he published Géométrie de position in which directed magnitudes were first systematically used in geometry.

Henri Cartan (1904–2008). French mathematician who worked on analytic functions, the theory of sheaves, homological theory, algebraic topology, and potential theory, producing significant developments in all these areas.

Pierre Cartier (1932–). French mathematician and member of Bourbaki. He stayed in the group until he retired in 1983. “I estimate that I contributed about 200 pages a year during all this time with Bourbaki.”

Mary Cartwright (1900–1998). British specialist in differential equations. She was the first woman to receive the Sylvester Medal and to serve on the Council of the Royal Society. She was president of the London Mathematical Society in 1961–1962, so far the only woman president. Cartwright had a gift for going to the heart of a matter and for seeing the important point, both in mathematics and in human affairs.

Augustin-Louis Cauchy (1789–1857). French mathematician who pioneered the study of analysis, both real and complex, and the theory of permutation groups. He also did research in convergence and divergence of infinite series, differential equations, determinants, probability, and mathematical physics.

Arthur Cayley (1821–1895). English mathematician at Cambridge University. His most important work was on the algebra of matrices and on non-Euclidean and n-dimensional geometry.

N. G. Chebotaryev (1894–1947). Russian mathematician who taught in Odessa and Kazan. His “density theorem” was used in Emil Artin’s solution of Hilbert’s 9th problem (the most general law of reciprocity.)

Pafnuty Lvovich Chebyshev (1821–1894). Russian mathematician known for his work in probability, statistics, and number theory. He is considered a founding father of Russian mathematics. Among his students were the prolific mathematicians Dmitry Grave, Aleksandr Korkin, Aleksandr Lyapunov, and Andrei Markov. According to the Mathematics Genealogy Project, Chebyshev has about 5000 mathematical “descendants.”

Shiing-Shen Chern (1911–2004). Chinese-American mathematician, who was one of the leaders in differential geometry in the 20th century.

Claude Chevalley (1909–1984). French mathematician who was a founding member of Bourbaki and a major influence on ring theory and group theory

William Schieffelin Claytor (1908–1967). U.S. mathematician. He worked as a researcher while teaching 18 to 21 hours per week and serving as chair of the department of mathematics at Howard University. In 1980 the National Association of Mathematicians instituted the Claytor Lecture Series in his honor.

Paul Cohen (1934–2007). U.S. mathematician who invented a new technique in set theory he called “forcing” and used it to prove the independence of the axiom of choice, and of the generalized continuum hypothesis. He spent most of his professional life at Stanford University.

Zerah Colburn (1804–1839). Famous U.S. child prodigy of the 19th century. Born in Cabot, Vermont, and educated at Westminster School in London, he was thought to be mentally retarded until the age of 7. When he was 7 years old he took 6 seconds to give the number of hours in 38 years, 2 months, and 7 days.

Jere Confrey. U.S. professor of mathematics education at Washington University. She was a cofounder of the UTEACH program for secondary math and science teacher preparation at the University of Texas in Austin and was the founder of the summer math program for young women at Mount Holyoke College and cofounder of Summer Math for Teachers.

John Horton Conway (1937–). English mathematician now at Princeton in the United States. He made major contributions to group theory, created a theory of “surreal numbers,” and has done leading research in knot theory, number theory, game theory, quadratic forms, coding theory, and tilings.

Richard Courant (1888–1972). German-American who was the leader of the Mathematics Institute at Göttingen. After that institute was virtually destroyed by Hitler, he went to New York and created at NYU an Institute of Mathematical Sciences which became a world leader in research. In 1964 it was named the Courant Institute after him.

Harold Scott MacDonald Coxeter (1907–2003). Canadian mathematician who graduated from Cambridge and worked most of his life in Canada. He worked mainly in geometry, making major contributions to the theory of polytopes, non-Euclidean geometry, group theory, and combinatorics.

Crafoord Prize. Science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife, Anna-Greta Crafoord. Administered by the Royal Swedish Academy of Sciences, the prize “is intended to promote international basic research in the disciplines of Astronomy and Mathematics; Geosciences; Biosciences, with particular emphasis on ecology and Polyarthritis (rheumatoid arthritis),” the disease from which Holger Crafoord severely suffered in his last years. The prize is presented by the King of Sweden (who also presents the awards at the December Nobel prize award ceremony). The prize sum of US $500,000 (2007) is intended to fund further research by the prize winner.

Mihaly Csikszentmihalyi (1934–). Hungarian-American psychology professor at Claremont Graduate University in Claremont, California. He is noted for his work on the study of creativity and subjective well-being and is best known as the architect of the notion of “flow” and for his years of research and writing on the topic.

George Dantzig (1914–2005). U.S. mathematics professor whose seminal work on the simplex method of linear programming is the foundation for much of systems engineering and is widely used in network design and component design in computer, mechanical, and electrical engineering.

Tobias Dantzig (1884–1956). Baltic-German-Russian-American mathematician who was the father of George Dantzig and the author of Number: The Language of Science, an outstanding book presenting deep mathematics in a way accessible to the layperson.

Joseph Dauben. U.S. historian of mathematics. He is the author of books about Georg Cantor and Abraham Robinson.

Harold Davenport (1907–1969). English mathematician known for extensive work on number theory. From about 1950 on he led a group that was the successor to the school of mathematical analysis of G. H. Hardy and J. E. Littlewood but more devoted to analytic number theory. This implied problem solving and hard analysis. The outstanding work in diophantine approximation of Klaus Roth and Alan Baker showed what this can do. This emphasis on concrete problems contrasted sharply with the abstraction of Bourbaki, which was then active just across the English Channel.

Chandler Davis (1926–). American-Canadian mathematician whose research has been principally on linear algebra and operator theory in Hilbert space. He began his writing career with Astounding Science Fiction in 1946. He has been a long-standing faculty member at the University of Toronto and is editor of The Mathematical Intelligencer.

Karel de Leeuw (1930–1978). U.S. mathematician at Stanford University who specialized in harmonic analysis and functional analysis. He received a doctorate from Princeton in 1954 under Emil Artin. He was murdered by Theodore Streleski, a Stanford doctoral student for 19 years whom he briefly advised.

Pierre Deligne (1944–). Belgian mathematician who was a student of Grothendieck who turned Grothendieck’s philosophy of motives into the driving force behind many areas of current algebraic geometry and arithmetic. Deligne has brought about a new understanding of the cohomology of varieties.

Jean Delsarte (1903–1968). French mathematician and member of Bourbaki who was best known for his work on mean periodic functions and translation operators. He was an analyst of great power and originality.

Abraham de Moivre (1667–1754). A French mathematician, famous for de Moivre’s formula, which links complex numbers and trigonometry, and for work on the normal distribution and probability theory. His parents were Protestants. Freedom of worship had been guaranteed in France since 1598 by the Edict of Nantes, but in 1685 Louis XIV revoked the Edict, leading to the expulsion of the Huguenots. De Moivre was imprisoned but eventually was able to emigrate to England. As a foreigner in England, he was never able to gain a university position, but lived in poverty on his earnings as a tutor. Nevertheless, he became a friend of Isaac Newton, Edmund Halley, and James Stirling, and was elected a Fellow of the Royal Society. Newton used to fetch him every evening for philosophical discourse from the coffee house where he could usually be found.

Augustus De Morgan (1806–1871). English mathematician and logician who defined and introduced the term “mathematical induction,” thereby putting that method on a rigorous basis.

René Descartes (1596–1650). French philosopher whose work is often considered the origin of modern philosophy. In his book La géométrie he introduced the systematic application of algebra to geometry, the first major advance in geometry since ancient times.

Jean Dieudonné (1906–1992). French mathematician who was a founding member of Bourbaki. He drafted much of the Bourbaki series of texts and also became provost of the University of Nice.

James A. Donaldson (1941–). U.S. mathematician and dean at Howard University who has worked on differential equations and applied mathematics.

Joseph Doob (1910–2004). U.S. probabilist who won the Steele Prize for his fundamental work on establishing probability as a branch of mathematics and for his continuing profound influence on its development.

Johann Peter Gustav Lejeune Dirichlet (1805–1859). German mathematician who proved in 1837 that, in any arithmetic progression whose first term has no common factor with the difference, there are infinitely many primes.

Underwood Dudley (1937–). U.S. mathematics professor at De Pauw University in the United States.

Freeman Dyson (1923–). English-American physicist and mathematician famous for his collaboration with Richard Feynman on Feynman’s method of path integrals. He is the author of many books on science and social problems.

Nikolai Vladimirovich Efimov (1910–1982). Russian geometer and administrator at Moscow State University.

Samuel Eilenberg (1913–1998). Polish-American mathematician who coauthored with Norman Steenrod the famous text Foundations of Algebraic Topology in 1952. At that time there were many different and confusing versions of homology theory. Eilenberg and Steenrod showed how to state all these different theories as functors of homology, mapping the category of pairs of spaces to the category of groups or rings and employing suitable axioms such as “excision.” He had a second life as a collector and connoisseur of modern art.

Albert Einstein (1879–1955). German physicist who contributed more than any other scientist to the modern vision of physical reality. His special and general theory of relativity is the most satisfactory model of the large-scale universe.

Paul Erdős (1913–1996). Hungarian mathematician who was one of the leading advocates and creators of discrete mathematics. He was peerless at both posing and solving problems in combinatorics, number theory, and other areas.

Alexander Sergeyevich Esenin-Volpin (1924–). Russian-American poet and mathematician. In the former Soviet Union, he was a leader of the human rights movement and spent 14 years incarcerated in prisons, in psikhushkas, and in exile.

Euclid of Alexandria (c. 325 BC–c. 265 BC). Euclid’s treatise on geometry, The Elements, dominated Western mathematics education for more than 2000 years.

Leonhard Euler (1707–1783). Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus, number theory, fluid dynamics, and elasticity.

Daniel Leonard Everett (1951–). U.S. linguistics professor best known for his study of the Amazon Basin’s Pirahã people and their language.

Gerd Faltings (1954–). German mathematician known for his work on arithmetic algebraic geometry. He was awarded the Fields Medal in 1986 for proving the Mordell conjecture, which states that any nonsingular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points.

Pierre de Fermat (1601–1665). French lawyer and government official remembered for his work on number theory, in particular on Fermat’s Last Theorem, and for important contributions to the foundations of the calculus and probability.

Fields Medal. Award given to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every 4 years. The Fields Medal is widely viewed as the top honor a mathematician can receive. It comes with a monetary award, which in 2006 was $15,000 (Canadian). It was founded at the behest of Canadian mathematician John Charles Fields.

Ludwik (Ludwig) Fleck (1896–1961). Polish medical doctor and biologist who developed the concept of thought collectives. This concept is important in the philosophy of science and the sociology of science. It helps to explain how scientific ideas change over time and is related to Thomas Kuhn’s later notion of paradigm shift. Fleck felt that the development of scientific insights is not unidirectional and does not consist of just accumulating new pieces of information but also in overthrowing old ones.

Catherine Fosnot. U.S. mathematics educator. Her main research is on the application of current models of cognitive psychology to the teaching of mathematics. Fosnot and her colleagues designed realistic problem situations as the starting point of investigation, inviting learners “to mathematize” initially in their own informal ways. Classrooms are thereby turned into workshops with learners engaged in inquiry, subsequently proving and communicating their thinking to their peers.

Jean Baptiste Joseph Fourier (1768–1830). French mathematical physicist who established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions.

Hans Freudenthal (1905–1990). Dutch mathematician who worked on the characters of the semisimple Lie groups between 1954 and 1956. Later he moved into the history of mathematics and mathematical education and had a great influence on the development of mathematics education research around the world.

Kurt Friedrichs (1901–1982). German-American mathematician who was coauthor with Richard Courant of the classic work Supersonic Flow and Shock Waves.

Ferdinand Georg Frobenius (1849–1917). German algebraist who was a professor at Berlin.

Dmitry Borisovich Fuchs (1939–). Russian-American mathematician who is a professor at the University of California, Davis, and prominent researcher in representations of infinite-dimensional Lie algebras. He was formerly a professor at Moscow State University and an instructor at the Jewish People’s University.

Galileo Galilei (1564–1642). Italian scientist who formulated the basic law of falling bodies, which he verified by careful measurements. He constructed a telescope with which he studied lunar craters, discovered four moons revolving around Jupiter, and espoused the Copernican cause.

Howard Gardner (1943–). U.S. cognitive psychologist at Harvard Graduate School of Education. He is best known for his theory of multiple intelligences. In 1981 he was awarded a MacArthur Prize Fellowship.

Carl Friedrich Gauss (1777–1855). German mathematician who worked in a wide variety of fields in both mathematics and physics, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. His work has had an immense influence in many areas.

David Hillel Gelernter (1955–). U.S. computer scientist who is a professor of computer science at Yale University. In the 1980s, he made seminal contributions to the field of parallel computation. In 1993 Gelernter was critically injured opening a mail bomb sent by Theodore Kaczynski, who at that time was an unidentified but violent opponent of technology, dubbed by the press as “the Unabomber.” Gelernter’s right hand and eye were permanently damaged. He chronicled the ordeal in his 1997 book Drawing Life: Surviving the Unabomber.

Israel Moiseevich Gelfand (1913–2009). Russian mathematician who was one of the most prolific and original mathematicians of the 20th century. He published over 500 papers in mathematics, applied mathematics, and biology. He was deeply involved in education, and established a correspondence school that brought rich mathematical experiences to students all over the Soviet Union.

Paulus Gerdes. Mozambique professor of mathematics at the Eduardo Mondlane University and at the Universidade Pedagogica in Mozambique for many years, serving as rector of the latter from 1989 to 1996. He is the director of the Ethnomathematics Research Centre and author of Geometry from Africa, Women, Art and Geometry in Southern Africa, and Culture and the Awakening of Geometrical Thinking.

Marie-Sophie Germain (1776–1831). French mathematician who made a major contribution to number theory, acoustics, and elasticity. She worked outside the established institutions that were closed to women.

Adele Gödel (1899–1981). Kurt Gödel’s wife, an Austrian who for many years nurtured and protected him. When she suffered an incapacitating stroke, he cared for her devotedly until she required emergency surgery and was hospitalized for nearly 6 months. Then Gödel had to fend for himself, and his fear of poisoning led to self-starvation. He died on January 14, 1978. At her death she bequeathed the rights to Gödel’s papers to the Institute for Advanced Study.

Kurt Gödel (1906–1978). Austrian philosopher and logician. His “incompleteness theorem” says that in any axiomatic mathematical system that includes the natural numbers, there are propositions that cannot be proved or disproved within the axioms of the system. At the Institute for Advanced Study he was a close friend of Albert Einstein.

Rebecca Goldstein (1950–). U.S. novelist and professor of philosophy. She has published six novels, a collection of stories, and two biographical studies—of mathematician Kurt Gödel and philosopher Baruch Spinoza.

Wayne Gould (1945–). Retired Hong Kong judge known for popularizing su do ku puzzles. Gould spent 6 years writing a computer program known as Pappocom Sudoku that mass-produces puzzles for the global market.

Evelyn Boyd Granville (1924–). Second African-American woman to earn a Ph.D. in mathematics. She was a student of Einar Hille at Yale. She worked as an applied mathematician for IBM and North American Aviation and taught at Fisk University in Nashville, at California State University in Los Angeles, and at Texas College in Tyler, Texas. Smith College, where she had earned her bachelor’s degree, awarded her an honorary doctorate in 1989.

Mary Gray (1939–). U.S. professor at American University in Washington, D.C. whose focus is on statistics. For years her name was virtually synonymous with the Association for Women in Mathematics. She is also a human rights activist.

John W. Green (1943–). U.S. mathematician who was a student of Robert Lee Moore. Green was on the University of Oklahoma faculty for 15 years. He then obtained a Ph.D. in mathematical statistics from Texas A. & M. University and then taught at the University of Delaware for 5 years. He is currently employed by E. I. DuPont as a senior research biostatistician.

Alexandre Grothendieck (1928–). German-born but later stateless mathematician who worked in France. From 1959 to 1970 a whole new school of mathematics flourished under Grothendieck’s leadership. His Séminaire de Géométrie Algébrique established the Insitut des Hautes Etudes Scientifiques (IHES) as a world center of algebraic geometry. He received the Fields Medal in 1966.

Jacques Salomon Hadamard (1865–1963). French mathematician who was a professor at the Sorbonne in Paris and proved the prime number theorem in 1896. This states that the number of primes < n is asymptotic to n/log n as n tends to infinity.

Hans Hahn (1879–1934). Austrian mathematician remembered for the Hahn-Banach theorem. He made important contributions to the calculus of variations, developing ideas of Weierstrass.

Paul Halmos (1916–2006). Hungarian-American mathematician well known for graduate texts on mathematics dealing with finite-dimensional vector spaces, measure theory, ergodic theory, and Hilbert space. Many of these books were the first systematic presentations of their subjects in English.

Israel Halperin (1911–2007). Canadian mathematician and human rights activist. He was a graduate student of John von Neumann at Princeton University. In February 1946, he was arrested and accused of espionage in Canada in connection with the defection of Igor Gouzenko, a Soviet cipher clerk. After some arduous questioning and confinement lasting several weeks, Halperin was eventually cleared. He was elected a Fellow of the Royal Society of Canada in 1953 and won the Henry Marshall Tory Medal in 1967. He authored more than 100 academic papers, was awarded an honorary doctorate of laws from Queen’s in 1989, and was made a Member of the Order of Canada for his humanitarian work and his mathematics.

George Bruce Halsted (1853–1922). U.S. mathematician. Eccentric and sometimes spectacular, Halsted became internationally known as a scholar, teacher, promoter, and popularizer of mathematics.

Richard Hamilton (1943–). U.S. mathematician who received a Ph.D. from Princeton University in 1966 under the direction of Robert Gunning. He is Professor of Mathematics at Columbia University and in 1996 was awarded the Oswald Veblen Prize of the American Mathematical Society.

Godfrey Harold Hardy (1877–1947). English mathematician who was a professor at Oxford and Cambridge, England. Hardy’s interests covered many topics of pure mathematics: diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes.

Jenny Harrison (1949–). U.S. professor of mathematics at the University of California at Berkeley. She developed a theory of quantum calculus that unifies an infinitesimal calculus with the classical theory of the smooth continuum. The methods apply to a class of domains called chainlets, which include soap films, fractals, graphs of L₁ functions, and charged particles, as well as smooth manifolds. Harrison’s lawsuit, based on sex discrimination in the Berkeley mathematics department’s tenure decision in 1987, attracted international attention.

Helmut Hasse (1898–1979). German mathematician who worked on algebraic number theory and was known for fundamental contributions to class field theory, the application of p-adic numbers to local class field theory and diophantine geometry (the Hasse principle) and to local zeta functions. He was Hermann Weyl’s replacement at Göttingen in 1934. After the defeat of Germany he returned to Göttingen briefly but was excluded by the British authorities. After brief appointments in Berlin, in 1948 he settled permanently as a professor in Hamburg.

Eduard Helly (1884–1943). Austrian mathematician who worked on functional analysis and in 1912 proved the Hahn-Banach theorem, 15 years before Hahn and 20 years before Banach.

Ravenna Helson. U.S. professor at the University of California, Berkeley, adjunct professor emeritus of the Department of Psychology of Mills College, and director of the Mills Longitudinal Study, which she initiated. Long interested in gender issues, she works with students on data from the Mills Study, which has followed about 120 women for 40 years, from age 21 to age 61. The Mills Study examines long-term personality components, social influences on personality, and processes of growth and development.

Claudia Henrion. U.S. author of Women in Mathematics: The Addition of Difference. She lives in New Hampshire in the United States.

Wilhelm Heydorn (1873–1958). German Protestant theologian, medical practitioner, and teacher. From 1921 to 1923 he studied at the University of Hamburg and worked until 1926 as medical practitioner. From 1926 to 1928 he studied to become a primary school teacher and worked from 1928 to 1933 as an assistant teacher. From 1934 to 1939 he and his wife, Dagmar, took care of Alexandre Grothendieck.

David Hilbert (1862–1943). German mathematician who was the leader of the famous Göttingen school of pure and applied mathematics. Hilbert’s work in geometry had the greatest influence in that area after Euclid. He made contributions in many areas of mathematics, physics, and logic.

Adolf Hurwitz (1859–1919). German mathematician who studied the genus of the Riemann surface and worked on how class number relations could be derived from modular equations.

Shokichi Iyanaga (1906–2006). Japanese mathematician who was a student of Teiji Takagi and contributed to algebraic number theory.

Fritz John (1910–1994). German-American mathematician who wrote classical papers on convexity, ill-posed problems, the numerical treatment of partial differential equations, quasi-isometry, and blow-up in nonlinear wave propagation.

Camille Jordan (1838–1922). French mathematician highly regarded for his work on algebra, group theory, and Galois theory.

Mark Kac (1914–1984). Polish-American mathematician who made major contributions to applied mathematics and probability theory. His sparkling juxtaposition of surprising phrases and constructions made his speech a delight to listen to.

Ted Kaczynski (1942–). U.S. mathematician (called “the Unabomber”) who gave up a Berkeley professorship to go live in the woods in Montana.

Nicholas M. Katz (1943–). U.S. mathematician working in algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. He played a significant role as a sounding board for Andrew Wiles when Wiles was developing in secret his proof of Fermat’s Last Theorem.

Felix Christian Klein (1849–1925). Very influential German mathematician. His Erlangen program defined a geometry as the properties of a space that are invariant under a given transformation group.

John R. Kline (1891–1955). U.S. professor of mathematics at the University of Pennsylvania from 1920 until his death in 1955. He was chair of the department from 1933 to 1954. He directed the Ph.D. theses of 13 students, including Dudley Weldon Woodard and W.W.S. Claytor, the second and third African-American mathematicians to earn a Ph.D.

Andrey Nikolaevich Kolmogorov (1903–1987). Russian mathemetics professor at the University of Moscow who became internationally famous for the rigorous development of measure-theoretic probability. He used this work to study the motion of the planets and the turbulent flow of air from a jet engine.

Sofia Vasilyevna Kovalevskaya (1850–1891). Russian mathematician, student, and friend of Karl Weierstrass, who made important contributions to the theory of differential equations.

Leopold Kronecker (1823–1891). German mathematician known for insisting that all mathematics must be based constructively on the natural numbers.

Ernst Kummer (1810–1893). German algebraist and number theorist famous for introducing “ideal numbers.”

Kazimierz Kuratowski (1896–1980). Polish mathematician who worked on topology and set theory. He is best known for his theorem giving a necessary and sufficient condition for a graph to be planar.

Imre Lakatos (1922–1974). Hungarian-British philosopher of mathematics and science at the London School of Economics who was influenced by Karl Popper and George Polya.

Edmund Georg Hermann Landau (1877–1938). German mathematician who gave the first systematic presentation of analytic number theory and wrote important works on the theory of analytic functions.

Pierre-Simon, Marquis de Laplace (1749–1827). French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics.

Jean Lave. U.S. social anthropologist whose research encompasses social practice in everyday mathematics, the study of apprenticeships, and learning communities. She is a professor at the University of California, Berkeley.

Anneli Lax (1922–1999). U.S. student of Richard Courant, professor at NYU, and editor of dozens of volumes of the New Mathematical Library.

Peter David Lax (1926–). Hungarian-American mathematician at the Courant Institute in New York. He is a leading researcher on partial differential equations and scattering theory and a winner of the Abel Prize (Norway) in 2005.

Solomon Lefschetz (1884–1972). U.S. mathematician who was the main source of work on the algebraic aspects of topology. He was a long-time chair at Princeton University.

Adrien-Marie Legendre (1752–1833). French mathematician who made important contributions to statistics, number theory, abstract algebra, and mathematical analysis.

Jean Leray (1906–1998). French mathematician who connected energy estimates for partial differential equations with fixed point theorems from algebraic topology in a highly original combination that cracked open the toughest problems. He was the first to adopt the modern viewpoint, where a function is thought of not as a complicated relation between two sets of variables but as a point in some infinite dimensional space.

Norman Levinson (1912–1975). U.S. Specialist in differential equations. He was considered the heart of mathematics at MIT, a man who combined high creative intellect with human compassion and dedication to science.

Roy Lisker (1938–). U.S. mathematician, novelist, musician, publisher, and social critic. Lisker publishes Ferment magazine and is the proprietor of the Ferment Press.

John Edensor Littlewood (1885–1977). Outstanding English mathematician. He was Hardy’s collaborator and worked on the theory of series, the Riemann zeta function, inequalities, and the theory of functions.

Nikolai Ivanovich Lobachevsky (1792–1856). Russian mathematician, at the University of Kazan, who in 1829 published his non-Euclidean geometry, the first account of the subject to appear in print.

Lee Lorch (1915–). U.S. mathematician and early civil rights activist who is currently professor emeritus at York University in Toronto, Canada. As a teacher at Black universities such as Fisk and Philander Smith, Lorch encouraged students, including Black women, to pursue graduate study in mathematics. Two colleges that fired him, Fisk University and City University, later awarded him honorary degrees. He was also honored by the U.S. National Academy of Sciences in 1990 and by Spelman College.

Edith Luchins (1921–2002). U.S. mathematician who taught at the Rensselaer Polytechnic Institute from 1962 until 1992. She was the first woman to be appointed a full professor at Rensselaer. Luchins’ research focused on mathematics and psychology. She worked on mathematical models of order effects in information processing; on gender differences in cognitive processes and their implications for teaching and learning mathematics; and on the roles of heuristics and algorithms in mathematical problem solving.

George Whitelaw Mackey (1916–2006). U.S. mathematician who joined the Harvard University mathematics department in 1943, was appointed Landon T. Clay Professor of Mathematics and Theoretical Science in 1969, and remained there until he retired in 1985. Mackey’s main research was in representation theory, ergodic theory, and related parts of functional analysis. Mackey did significant work on the duality theory of locally convex spaces, which provided tools for subsequent work in this area, including Alexandre Grothendieck’s work on topological tensor products.

MacTutor History of Mathematics. An extensive, searchable online archive of persons and concepts, written and edited by J. J. O’Connor and E. F. Robertson at the University of St. Andrews.

Wilhelm Magnus (1907–1990). German-American mathematician. In addition to his main research on group theory and special functions, he worked on problems in mathematical physics, including electromagnetic theory and applications of the wave equation. He was an outstanding supervisor of doctoral students at the Courant Institute, supervising 61 dissertations during his career.

Vivienne Malone-Mayes (1932–1995). U.S. professor who earned a B.A. (1952) and M.A. (1954) in mathematics at Fisk University. Dr. Mayes became an outstanding teacher as the first Black faculty member at Baylor University, which had rejected her as a student with an explicit antiblack policy 5 years earlier.

Benoit Mandelbrot (1924–). French-American mathematician who is a major contributor to the growing field of fractal geometry. He showed that fractals occur in many different places, in mathematics and elsewhere in nature.

José Luis Massera (1915–2002). Uruguayan mathematician. A theorem that bears his name solves the problem of the stability of nonlinear differential equations in terms of the Lyapunov exponents. Massera’s political activity resulted in his arrest on October 22, 1975. A broad and vigorous international campaign constantly demanded his release, and he regained freedom in March 1983.

Stanislaw Mazur (1905–1981). Polish mathematician who made important contributions to geometrical methods in linear and nonlinear functional analysis and to the study of Banach algebras. Stan Ulam recounts how Mazur gave the first examples of infinite games in the Scottish Café in Lvov.

John Milnor (1931–). U.S. mathematician. His most remarkable achievement, which played a major role in his winning the Fields Medal, was his proof that a seven-dimensional sphere can have several differential structures. This work opened up the new field of differential topology.

Hermann Minkowski (1864–1909). German mathematician who was a friend and collaborator of Hilbert. He developed a new view of space and time and laid the mathematical foundation of the theory of relativity.

Magnus Gösta Mittag-Leffler (1846–1927). Swedish mathematician who was a student of Weierstrass. He worked on the general theory of functions. His best known work concerned the analytic representation of a one-valued function.

Edwin Evariste Moise (1918–1998). U.S. mathematician. He was a topologist, a student of Robert Lee Moore. He helped decipher German and Japanese military signals in World War II. At Michigan he did important work on 3-manifolds, culminating in his proof that every 3-manifold can be triangulated.

Gaspard Monge (1746–1818). French mathematician considered the father of differential geometry because of his theory of curvature of a surface in 3-space.

Calvin C. Moore. U.S. functional analyst who has had a long career as professor and administrator at the University of California, Berkeley.

Robert Lee Moore (1882–1974). U.S. mathematician known for his work on general topology and the Moore method of teaching university mathematics.

Cathleen Morawetz (1923–). U.S. mathematician who was the second female president of the American Mathematical Society in 1995–1996. In 1998 she was awarded the National Medal of Science for pioneering advances in partial differential equations and wave propagation resulting in applications to aerodynamics, acoustics, and optics.

Louis Joel Mordell (1888–1972). British-American mathematician best known for his investigations of equations of the form y² = x³ + k, which had been studied by Fermat. His conjecture about rational points on algebraic curves, which was proved in 1983 by Gerd Faltings, was an important ingredient in Andrew Wiles’ proof of Fermat’s Last Theorem.

Oskar Morgenstern (1902–1977). Austrian-American economist who, working with John von Neumann, helped found the mathematical field of game theory.

Harold Calvin Marston Morse (1892–1977). U.S. mathematician who developed variational theory in the large with applications to equilibrium problems in mathematical physics, a theory that is now called Morse theory and forms a vital role in the mathematics of global analysis.

Jürgen Moser (1928–1999). German mathematician who worked at the Courant Institute in New York and at the ETH in Geneva. He specialized in ordinary differential equations, partial differential equations, spectral theory, celestial mechanics, and stability theory. The leading theme of virtually all of Moser’s work in dynamics is the search for elements of stable behavior in dynamical systems with respect to either initial conditions or perturbations of the system.

Robert Parris Moses (1935–). U.S. Harvard-trained educator (known as Bob Moses) who was a leader in the civil rights movement and later founded the nationwide U.S. Algebra Project.

Gilbert Murray (1948–1995). Former president of the California Forestry Association, a timber industry lobbying group. In 1995 he was killed by a bomb mailed by “the Unabomber,” Ted Kaczynski. The bomb was addressed to the CFA’s previous president, Bill Dennison.

John Nash (1928–). U.S. mathematician. In 1949, while studying for his doctorate, he wrote a paper which 45 years later won a Nobel Prize for economics. P. Ordeshook wrote: “The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory.”

Rolf Nevanlinna (1895–1980). Finnish mathematician whose most important work was on harmonic measure, which he invented. He also developed the theory of value distribution named after him. The main results of Nevanlinna theory appeared in 1925 in a 100-page paper that Weyl called “one of the few great mathematical events of our century.”

Isaac Newton (1643–1727). English mathematician and physicist who laid the foundation for differential and integral calculus. His work on optics and gravitation makes him one of the greatest scientists the world has known.

Jerzy Neyman (1894–1981). Polish-American statistician who put forward the theory of confidence intervals, which plays a central role in statistical theory and data analysis. His contribution to the theory of contagious distributions is still of great utility in the interpretation of biological data.

Nel Noddings (1929–). American feminist and philosopher best known for her work on the philosophy of education, educational theory, and the ethics of care.

Emmy Noether (1882–1935). German mathematician who was the principal founder of modern abstract algebra. She is particularly known for her study of chain conditions on ideals of rings. To this day, she is an inspiration to women mathematicians.

Sergei Novikov (1938–). Russian mathematician and Fields medalist. Both his parents were famous mathematicians. Novikov studied at the Faculty of Mathematics and Mechanics of Moscow University (1955–1960) and has worked there since 1964 in the department of differential geometry. His work has played an important part in building a bridge between modern mathematics and theoretical physics.

King Oscar II (1829–1907). Successor to his brother, Carl IV, as King of Norway and Sweden in 1872. He married Sophia of Nassau, and their eldest son, Gustav, became King of Sweden. Oscar II was the last king of the Norwegian-Swedish union. Oscar supported mathematical research in Sweden, and an important prize was named after him.

Blaise Pascal (1623–1662). Influential French mathematician and philosopher who contributed to many areas of mathematics. He worked on conic sections and projective geometry, and in correspondence with Fermat he laid the foundations for the theory of probability.

John Allen Paulos (1945–). U.S. professor of mathematics at Temple University in Philadelphia who has written on the importance of mathematical literacy and the mathematical basis of humor.

Sir Roger Penrose (1931–). English mathematical physicist renowned for contributions to general relativity and cosmology. He is also a recreational mathematician and philosopher. Penrose proved that, under certain conditions, in a gravitational collapse space-time cannot be continued and classical general relativity breaks down. He looked for a unified theory combining relativity and quantum theory since quantum effects become dominant at the singularity. One of Penrose’s major breakthroughs was his twistor theory, an attempt to unite relativity and quantum theory.

Grigori Yakovlevich Perelman (1966–). Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston’s geometrization conjecture. This solves in the affirmative the famous Poincaré conjecture posed in 1904 and regarded as one of the most important and difficult open problems in mathematics. In August 2006, Perelman was awarded the Fields Medal for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” However, he declined to accept the award or to appear at the congress.

Rózsa Péter (1905–1977). Hungarian mathematician best known for her work on recursion theory. She attended Eötvös Loránd University, where she received a Ph.D. in 1935. During Miklós Horthy’s collaboration with Nazi Germany, she was forbidden to teach because of her Jewish origin. After the war she published her key work, Recursive Functions. She taught at her alma mater from 1955 until her retirement in 1975.

Ivan Georgievich Petrovsky (1901–1973). Russian mathematician in the field of partial differential equations. He greatly contributed to the solution of Hilbert’s 19th and 16th problems. He also worked on boundary value problems, probability, and topology of algebraic curves and surfaces. Among his students were Olga Ladyzhenskaya and Olga Oleinik. Petrovsky was the rector of Moscow State University (1951–1973). and the head of the International Congress of Mathematicians (Moscow, 1966). He regarded his rectorship of Moscow State University as the most important thing in his life, even more important than his mathematical research.

Wolodymyr V. Petryshyn (1929–). U.S. mathematician whose major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.

Jean Piaget (1896–1980). Swiss developmental psychologist well known for his work studying children, his theory of cognitive development, and his theory of knowledge acquisition called “genetic epistemology.” He was one of the major figures of 20th century psychology. In 1955 he created the International Centre for Genetic Epistemology in Geneva and directed it until 1980.

Vera Pless (1931–). U.S. mathematician who worked at the Air Force Cambridge Research Laboratory from 1963 to 1972, where she became a leading expert on coding theory. In 1963 she published power moment identities on weight distributions in error-correcting codes. These are used to determine the complete weight distributions of several quadratic residue codes.

Jules Henri Poincaré (1854–1912). French mathematician who was one of the 20th century giants in this field. He was the originator of algebraic topology and of the theory of analytic functions of several complex variables.

Siméon-Denis Poisson (1781–1840). French mathematician, geometer, and physicist.

George Polya (1887–1985). Hungarian-American mathematician who worked on probability, analysis, number theory, geometry, combinatorics, and mathematical physics. His writings on heuristics and problems solving are very influential.

Jean-Victor Poncelet (1788–1867). French mathematician who was one of the founders of modern projective geometry. His development of the pole and polar lines associated with conics led to the principle of duality.

Lev Pontryagin (1908–1988). Russian mathematician who constructed a general theory of characters for commutative topological groups. He used this theory of characters to prove that any Abelian locally Euclidean topological group can be given the structure of a Lie group (Hilbert’s 5th problem for the case of Abelian groups).

Karl Popper (1902–1994). Austrian who was a major philosopher of science and opposed all forms of skepticism, conventionalism, and relativism. In human affairs he was a committed advocate and defender of an “open society.”

Marian Boykan Pour-El (?–2009). U.S. logician and mathematician who was the author of many articles on logic and its applications in mathematics and physics. She specialized in computability and noncomputability of mathematical and physical theories.

Tibor Radó (1895–1965). Hungarian-American mathematician. The Plateau problem was studied by Plateau, Weierstrass, Riemann, and Schwarz but was finally solved by Douglas and Radó. He used conformal mappings of polyhedra, applying a limit theorem to certain approximations to obtain the minimal surface required. The solution did not exclude the possibility that the minimal surface might contain a singularity. In fact, it never does contain a singularity; this was shown for the first time by Osserman in 1970.

George Yuri Rainich (1886–1968). Russian-American mathematical physicist who was well-known in the early 20th century. Rainich’s research centered on general relativity and early work toward a unified field theory.

Srinivasa Aiyangar Ramanujan (1887–1920). Indian mathematician who became well known through his close collaboration with G. H. Hardy. Ramanujan made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Lord Rayleigh (John William Strutt) (1842–1919). English physicist whose theory of scattering, published in 1871, was the first correct explanation of why the sky is blue. The first volume of his major text, The Theory of Sound, on the mechanics of a vibrating medium, was published in 1877, and the second volume, on acoustic wave propagation, was published the following year.

Constance Bowman Reid (1918–). U.S. author of several biographies of mathematicians and popular books about mathematics. She is the sister of mathematician Julia Robinson.

Alfred Rényi (1921–1970). Hungarian mathematician. He obtained a Ph.D. at Szeged under F. Riesz, was taught by Fejér at Budapest, and then went to Russia to work with Linnik on the theory of numbers, in particular on the Goldbach conjecture. He discovered methods described by Turán as among the strongest methods of analytical number theory. After returning to Hungary he worked on probability. He published joint work with ErdŐs on random graphs and also considered random space filling curves. He is remembered for proving that every even integer is the sum of a prime and an almost prime number (one with only two prime factors).

Bernhard Riemann (1826–1866). German mathematician who was a student of Gauss and a professor at Göttingen. His ideas concerning the geometry of space had a profound effect on the development of modern theoretical physics. He clarified the notion of an integral by defining what we now call the Riemann integral.

Frigyes Riesz (1880–1956). Hungarian mathematician who was a founder of functional analysis and whose work has many important applications in physics. He was the leader of the mathematical school at Szeged.

Herbert Ellis Robbins (1915–2001). U.S. mathematician and statistician who did research on topology, measure theory, statistics, and a variety of other fields. He was the coauthor, with Richard Courant, of What Is Mathematics?, a popularization that is still (as of 2009) in print. The Robbins lemma, used in empirical Bayes methods, is named after him.

Abraham Robinson (1918–1974). Israeli-American mathematician whose most famous invention was nonstandard analysis which he introduced in 1961. A nonstandard model for the system of real numbers has the feature of being a non-Archimedean totally ordered field that contains a copy of the real number system.

Julia Hall Bowman Robinson (1919–1985). U.S. mathematician who worked on computability, decision problems, and nonstandard models of arithmetic. The recipient of many honors, Robinson was the first female mathematician member of the National Academy of Sciences and the first female president of the American Mathematical Society. She also received a McArthur grant.

Raphael Robinson (1911–1995). U.S. professor of mathematics at Berkeley, who worked on complex analysis, logic, set theory, geometry, number theory, and combinatorics.

Judith Roitman (1945–). U.S. mathematician who is currently a professor at the University of Kansas. She specializes in applications of set theory to topology and Boolean algebra.

Gian-Carlo Rota (1932–1999). Italian-American professor at MIT in applied mathematics and philosophy. He began as a functional analyst and moved on to combinatorics, where he became a leading national and international figure. He was also a person of great cultural and literary attainment. He was a mathematical gourmet.

Henry Roth (1906–1995). U.S. author whose best-known work is Call It Sleep (1934), a classic in Jewish-American literature.

Mary Ellen (Estill) Rudin (1924–). U.S. mathematician. In 1971 the University of Wisconsin promoted her from lecturer to full professor. “Nobody even asked me if I wanted to be a professor. I was simply presented with this full professorship.” Her research has been in set-theoretic topology, especially using axiomatic set theory. Methods from general topology unexpectedly solved several difficult problems in functional analysis and in geometric and algebraic topology. Two developments revolutionized the field: the creation of infinite-dimensional topology and set-theoretic topology. It has been mainly due to Dick Anderson and Mary Ellen Rudin that these fields have dominated general topology ever since.

Carle David Tolmé Runge (1856–1927). German mathematician who worked on a procedure for the numerical solution of algebraic equations and later studied the wavelengths of the spectral lines of elements.

Bertrand Arthur William Russell (1872–1970). English mathematician and philosopher who published a vast number of books on logic, theory of knowledge, and many other topics. His best known work was Principia Mathematica.

Oliver Sacks (1933–). U.S.-based British neurologist who has written popular books about brain pathologies in his patients, the most famous of which is Awakenings.

Pierre Samuel (1921–2009). French mathematician who worked with Alexandre Grothendieck’s environmentalist movement and was a cofounder of the French Green Party. He was known for work in commutative algebra and its applications to algebraic geometry. The two-volume work Commutative Algebra that he wrote with Oscar Zariski is a classic.

Jane Cronin Scanlon (1922–). U.S. mathematician who began her college studies in physics but later become more interested in the abstract nature of mathematics. She received a Ph.D. in mathematics in 1949 from the University of Michigan. Her research interests are in mathematical biology and partial differential equations.

Alice Schafer (1915–2009). U.S. mathematician. She lists two specializations: abstract algebra (group theory) and women in mathematics. Through her work as a mathematics educator, she became a champion for full participation of women in mathematics. From 1962 to 1980 she was professor and head of the mathematics department at Wellesley College.

Laurent Schwartz (1915–2002). French mathematician whose paper on the theory of distributions is one of the classical mathematical papers of our time. Later he worked on stochastic differential calculus. He campaigned against American involvement in Vietnam, the Soviet invasion of Afghanistan, and the Russian war against Chechnya. He also was an avid collector of butterflies, with over 20,000 specimens. Schwartz wrote in his autobiography, “To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.”

Atle Selberg (1917–2007). Norwegian mathematician who was awarded a Fields Medal for his work on generalizations of the sieve methods of Viggo Brun, and for his major work on the zeros of the Riemann zeta function, where he proved that a positive proportion of its zeros satisfy the Riemann hypothesis. Selberg is also well known for his elementary proof of the prime number theorem, which says that the number of primes ≤ n is asymptotic to n/log n as n tends to infinity.

Jean-Pierre Serre (1926–). French mathematician who is one of the leading mathematicians of the 20th century, active in algebraic geometry, number theory, and topology. His early work was on spectral sequences. The Serre spectral sequence provided a tool to work effectively with the homology of fiberings. For this work on spectral sequences and his work developing complex variable theory in terms of sheaves, Serre was awarded a Fields Medal at the International Congress of Mathematicians in 1954. Serre’s theorem led to rapid progress not only in homotopy theory but also in algebraic topology and homological algebra in general.

Igor Rostislavovich Shafarevich (1923–). Russian mathematician who was the founder of the major school of algebraic number theory and algebraic geometry in the former Soviet Union. He made major contributions to the inverse problem of Galois theory as well as to class field theory, settling some long-outstanding conjectures. More recently he has made important advances in algebraic geometry. He was an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov’s human rights committee from 1970. In 1972, he joined the group of dissidents led by Solzhenitsyn. As a consequence, he was dismissed from the University of Moscow in 1975. In 1989 he published a book, Russophobia, which contained surprising anti-Semitic slanders.

Allen Lowell Shields (1927–1989). U.S. mathematician who worked on a wide range of mathematical topics including measure theory, complex functions, functional analysis, and operator theory.

Isadore Singer (1924–). U.S. mathematician famous for his deep and spectacular work on geometry, analysis, and topology. Singer’s five papers with Michael F. Atiyah on the index theorem for elliptic operators and his three papers with Atiyah and V. K. Patodi on the index theorem for manifolds with boundary are among the great classics of global analysis. They have spawned many developments in differential geometry, differential topology, and analysis.

Steve Smale (1930–). U.S. mathematician who began his career as an instructor at the University of Chicago. In 1958 he astounded the mathematical world with a proof of a sphere eversion. He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5. He later generalized the ideas in a 107-page paper that established the h-cobordism theorem. After making great strides in topology, he then turned to the study of dynamical systems. His first contribution was the Smale horseshoe, which jump-started significant research in dynamical systems. He also outlined a research program carried out by many others. Smale is also known for injecting Morse theory into mathematical economics, as well as recent explorations of various theories of computation. He is also known as world-class collector and photographer of mineral specimens.

Alexei B. Sossinsky (1937–). Russian senior researcher in the laboratory of mathematical methods of the Institute for Problems in Mechanics at the Russian Academy of Sciences, and professor at the Higher Mathematics College of the Independent University of Moscow.

Julian Cecil Stanley (1918–2005). U.S. psychologist, educator, and advocate of accelerated education for academically gifted children. He founded the Johns Hopkins University Center for Talented Youth (CTY), as well as a related research project, the Study of Mathematically Precocious Youth, which in 2005 was renamed the Julian C. Stanley Study of Exceptional Talent (SET). Stanley was also widely known for his book, coauthored with Donald Campbell, Experimental and Quasi-experimental Designs for Research.

Clarence F. Stephens (1917–). U.S. mathematician who graduated from Johnson C. Smith University in 1938 with a B.S. degree in mathematics. He received an M.S. (1939) and a Ph.D. (1943) from the University of Michigan. In 1947 he joined the mathematics faculty at Morgan State University. Stephens remained at Morgan State until 1962, at which time he accepted an appointment as professor of mathematics at SUNY Geneseo. In 1969 he left Geneseo to join the mathematics faculty at SUNY Potsdam, where he served as chair of the mathematics department until his retirement in 1987. During Stephens’ tenure at SUNY Potsdam, the department became nationally known as a model of teaching excellence in mathematics.

Marshall Harvey Stone (1903–1989). U.S. mathematician who was a noted chairman at the University of Chicago. He is best known for the Stone-Weierstrass theorem on uniform approximation of continuous functions by polynomials.

Ted Streleski (1936–). U.S. graduate student in mathematics at Stanford University who murdered professor Karel de Leeuw with a ball peen hammer. Streleski felt the murder was justifiable because de Leeuw had withheld departmental awards from him and demeaned him in front of his peers. Streleski had been pursuing a doctorate in the mathematics department for 19 years. He served 7 years in prison and was eligible for parole on three occasions but turned it down because the conditions of parole required him to not set foot on the Stanford campus. Upon release he said, “I have no intention of killing again. On the other hand, I cannot predict the future.”

Steven H. Strogatz (1959–). U.S. mathematician and Jacob Gould Schurman Professor of Applied Mathematics at Cornell University who is known for contributions to the study of synchronization in dynamical systems and for work in a variety of areas of applied mathematics, including mathematical biology and complex network theory.

Dirk Jan Struik (1894–2000). Dutch-American mathematician and Marxist theoretician who spent most of his life in the United States as a professor at MIT. He was a specialist in differential geometry.

Beauregard Stubblefield (1923–). U.S. mathematician who from 1952 to 1956 was head of the department of mathematics of the University of Liberia at Monrovia. From 1959 to 1960 he served as lecturer and National Science Foundation Post-Doctoral Fellow at the University of Michigan in Ann Arbor, Michigan. Since then he has been a faculty member at Stevens Institute of Technology in Hoboken, New Jersey, at Oakland University in Rochester, Michigan, at Texas Southern University, and at Appalachian State University in Boone, North Carolina.

Bella Abramovna Subbotovskaya (1938–1982). Russian mathematician who was the founder of the Jewish People’s University in Moscow from 1978 to 1983.

James Joseph Sylvester (1814–1897). English mathematician who did important work on matrix theory. In 1851 Sylvester discovered the discriminant of a cubic equation and first used the name “discriminant” for equations of higher order. He founded the first U.S. graduate department of mathematics at Johns Hopkins, and the American Journal of Mathematics.

John Lighton Synge (1897–1995). Irish-Canadian professor who made outstanding contributions to classical mechanics, geometrical mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical networks, mathematical methods, differential geometry and, above all, Einstein’s theory of relativity.

Gábor Szegö (1895–1985). Hungarian-American mathematician who worked on extremal problems and Toeplitz matrices. He was a professor and chair of mathematics at Stanford University and a longtime collaborator of George Polya.

Teiji Takagi (1875–1960). Japanese mathematician who was a student of Hilbert. He worked on class field theory, building on Heinrich Weber’s work.

Yutaka Taniyama (1927–1958). Japanese mathematician who posed two problems at the Symposium on Algebraic Number Theory held in Tokyo in 1955 that form the basis of the Shimura-Taniyama conjecture: “Every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field.” This conjecture was a major factor in Wiles’ proof of Fermat’s Last Theorem.

Olga Taussky-Todd (1906–1995). Distinguished and prolific Austrian-American mathematician who wrote about 300 papers. Her best known and most influential work was in matrix theory, and she also made important contributions to number theory.

Jean E. Taylor (1944–). U.S. mathematician who went to the University of California to study chemistry but audited S. S. Chern’s class in differential geometry, which inspired her. With his help, she transferred to mathematics She then earned a M.Sc. in mathematics at Warwick and a Ph.D. at Princeton. In 1973 she went to Rutgers and rose to the rank of professor. In 2002 she retired and settled into the Courant Institute. Her research has been primarily in the field of geometric measure theory applied to problems of optimal shapes of crystals, both in equilibrium and otherwise.

Richard Taylor (1962–). English mathematician who received his Ph.D. from Princeton University in 1988. From 1995 to 1996 he held the Savilian Chair of Geometry at Oxford University and is currently the Herschel Smith Professor of Mathematics at Harvard University. One of the two papers containing the published proof of Fermat’s Last Theorem is a joint work of Taylor and Andrew Wiles. In subsequent work, Taylor (along with Michael Harris) proved the local Langlands conjectures for GL(n) over a number field. Taylor, along with Christophe Breuil, Brian Conrad, and Fred Diamond, completed the proof of the Taniyama-Shimura conjecture.

William Thurston (1946–). U.S. professor at Cornell whose ideas revolutionized the study of topology in four dimensions and brought about a new interplay between analysis, topology, and geometry. This method is a new level of geometrical analysis—in the sense of powerful geometrical estimation on the one hand and spatial visualization and imagination on the other.

John Todd (1911–2007). English mathematician who spent 10 years in Washington at the National Applied Mathematical Laboratories, developing high-speed computer programming and becoming a world leader in numerical analysis and numerical algebra. In 1957 Todd and Olga Taussky moved to Cal Tech, where Todd developed the first undergraduate courses in numerical analysis and numerical algebra, prerequisites to learning computing.

Pál Turán (1910–1976). Hungarian mathematician. The most important and original of Turán’s results are in his power sum method and its applications. They led to interesting deep problems of a completely new type; they have quite unexpectedly surprising consequences in many branches of mathematics—differential equations, numerical algebra, and various branches of function theory.

Yoshisuke Ueda (1936–). Japanese mathematician and engineer who discovered chaos in 1961 in the course of studying certain differential equations with the use of analog computers.

Stanislaw Marcin Ulam (1909–1984). Polish-American mathematician who was long associated with the Los Alamos National Laboratory, where he solved mathematically the problem of how to initiate fusion in the hydrogen bomb. He also promoted the Monte Carlo method widely used in solving mathematical problems using statistical sampling.

Kristin Umland. U.S. professor of mathematics at the University of New Mexico. Her research focuses on the cognitive aspects of learning mathematics.

Georges Valiron (1884–1955). French mathematician who was secretary of the assembly meeting of the International Mathematical Union in 1932.

Bartel Leendert van der Waerden (1903–1996). Dutch algebraist and historian who was the author of a very influential textbook on abstract algebra.

Harry Schultz Vandiver (1882–1973). U.S. mathematician. At an early age he went to work as a customs house broker for his father’s firm. He never graduated from high school. When he was 18 he began to solve number theory problems in the American Mathematical Monthly. G. D. Birkhoff persuaded Vandiver to accept a post at Cornell University in 1919. In 1924 he moved to the University of Texas, being promoted to full professor in 1925 and then to distinguished professor of applied mathematics and astronomy in 1947. He retired in 1966 at the age of 84. He continued to work on extending Kummer’s methods of studying Fermat’s Last Theorem for increasingly large exponents. With hand calculations and help from his students, he had shown the result to be true for all n up to 600. In 1952 he was able to implement his methods on early computers and to prove the theorem true for all primes less than 2000.

Srinivasa Varadhan (1940–). Indian-American probabilist, professor at the Courant Institute at NYU, and winner of the Abel Prize in 2007 for fundamental contributions to probability theory and in particular for creating a unified theory of large deviation.

Oswald Veblen (1880–1960). U.S. mathematician and Princeton professor who made important contributions to projective and differential geometry, and topology.

William Spencer Vickrey (1914–1996). Columbia University economics professor whose Nobel Memorial Prize in Economics was announced just 3 days before he died. He authored the seminal 1961 Journal of Finance paper, “Counterspeculation, Auctions and Competitive Sealed Tenders,” which was the first instance of an economist using the tools of game theory to understand auctions.

Theodor von Kármán (1881–1963). Hungarian-American mathematician who founded the U.S. Institute of Aeronautical Sciences, continuing his research on fluid mechanics, turbulence theory, and supersonic flight. He studied applications of mathematics to engineering, aircraft structures, and soil erosion.

Niels Fabian Helge von Koch (1870–1924). Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake. He was born into a family of Swedish nobility. His father Richert Vogt von Koch (1838–1913) was a lieutenant colonel in the Royal Horse Guards of Sweden. Von Koch wrote several papers on number theory. One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem.

John von Neumann (1903–1957). Hungarian-American professor at the Institute for Advanced Study in Princeton. He built a solid framework for quantum mechanics. He also worked in game theory, studied what are now called von Neumann algebras, and was one of the pioneers of computer science.

Lev Semenovich Vygotsky (1896–1934). Soviet developmental psychologist and the founder of cultural-historical psychology. He graduated from Moscow State University in 1917. While at the Institute of Psychology in Moscow (1924–1934), he worked extensively on ideas about cognitive development, particularly the relationship between language and thinking. His writings emphasized the roles of historical, cultural, and social factors in cognition and language. Vygotsky died of tuberculosis in 1934, leaving a wealth of work that is still being explored.

Janice Anita Brown Walker (1949–). U.S. mathematician who earned her Ph.D. at Michigan in 1982. The atmosphere in the mathematics department, the support of many faculty members, and the camaraderie among the students made her time at Michigan rewarding, stimulating, and comfortable. Since 1992 she has been chair of the Mathematics and Computer Science Department at Xavier University in Cincinnati, Ohio.

Valerie Walkerdine. English specialist on gender and class and its impact on mathematical development. She teaches at the School of Social Science at the University of Cardiff in Wales.

Karl Theodor Wilhelm Weierstrass (1815–1897). German mathematician who was a very influential professor at Berlin. He was best known for his construction of the theory of complex functions by means of power series.

André Weil (1906–1998). French mathematician and a founding member of Bourbaki. He started the rapid advance of algebraic geometry and number theory by laying the foundations for abstract algebraic geometry and the modern theory of Abelian varieties. His work on algebraic curves has influenced a wide variety of areas, including some outside mathematics such as elementary particle physics and string theory.

Tilla Milnor Weinstein (d. 2002). U.S. geometer specializing in Lorentz surfaces. She was a professor and then chair of mathematics at Douglass College, Rutgers University.

Anna Pell Wheeler (1883–1966). U.S. mathematician who was the first female mathematician to address the American Mathematical Society. She fostered women’s participation in mathematics. Teaching longest at Bryn Mawr (1918–1948), where she also chaired the mathematics department, she specialized in integral equations. She was also an avid bird watcher and wildflower enthusiast.

Sylvia Wiegand (1945–). U.S. mathematician. Several members of her family were mathematicians. She wrote about her grandparents, Grace Chisholm Young and William Henry Young. Her interest in mathematics developed at a young age in response to the mathematical puzzles posed by her father, Laurence Chisholm Young, a mathematics professor at the University of Wisconsin. She received a Ph.D. in 1972 with a thesis in commutative algebra. She has co-authored many research papers with her husband, Roger Wiegand. They both hold appointments at the University of Nebraska.

Leo Wiener (1862–1939). U.S. professor of languages. Norbert Wiener’s father, Leo, attended medical school at the University of Warsaw but then went to Berlin where he began training as an engineer. He gave this up to emigrate to America. Arriving in New Orleans in 1880, Leo tried various jobs in factories and farms before becoming a school teacher in Kansas City. He progressed to the post of professor of modern languages at the University of Missouri and then left Missouri for Boston. He was appointed an instructor in Slavic languages at Harvard, but this did not pay enough to provide for his family and he kept other positions to augment his salary. He remained at Harvard and eventually was promoted to professor.

Norbert Wiener (1894–1964). U.S. mathematics professor at MIT. Best known for his very influential work on Brownian motion. He introduced a measure on the space of one-dimensional paths which brings in probability concepts in a natural way. From 1923 he investigated Dirichlet’s problem, producing work that had a major influence on potential theory.

Eugene Wigner (1902–1995). Hungarian-American physicist whose investigations of symmetry principles in physics are important far beyond nuclear physics proper. His methods and results have become an indispensable guide for the interpretation of the rich and complicated picture that has emerged from recent years’ experimental research on elementary particles.

Raymond L. Wilder (1896–1982). U.S. mathematician who received a Ph.D. from the University of Texas in 1923. The bulk of his academic career was at the University of Michigan. Wilder was one of the developers of algebraic topology. He was the author of several books on the philosophy of mathematics and the culture of mathematics.

Andrew John Wiles (1953–). English number theorist and professor at Princeton University. In 1995 Wiles proved Fermat’s Last Theorem, which states that if aⁿ + bⁿ = cⁿ, where a, b, c, and n are all positive integers, then n is less than or equal to 2.

Ellen Winner. U.S. professor of psychology at Boston College. Her work in the visual arts has focused on children’s sensitivity to aesthetic aspects of works of art, such as line quality, expression, and composition. Her most recent book explores misconceptions about children who are extremely gifted and makes a set of recommendations about how such children should be educated.

Melanie Matchett Wood (1981–). U.S. mathematician who, while a high school student in Indianapolis, became the first female to make the U.S. International Mathematical Olympiad Team, receiving silver medals in the 1998 and 1999 competitions. She attended Duke University, where she was named a Putnam Fellow. During the 2003–2004 year she studied at Cambridge University, where she won the Morgan Prize for work in Belyi-extending maps and in P-orderings. She was named the deputy leader of the U.S. team that finished second overall at the 2005 International Mathematical Olympiad.

Dudley Weldon Woodard (1881–1965). African-American mathematician who attended Wilberforce College in Ohio, receiving a bachelor degree (A.B.) in mathematics in 1903. He then received a B.S. degree in 1906 and an M.S. degree in mathematics from the University of Chicago in 1907. From 1907 to 1914, he taught mathematics at Tuskegee Institute and then moved to join the Wilberforce faculty from 1914 to 1920. In 1921 he joined the mathematics faculty at Howard University. He received a Ph.D. in mathematics in 1928 at the University of Pennsylvania. While at Howard, he was also selected dean of the College of Arts and Sciences.

John Worrall (1946–). English philosopher who studied for a Ph.D. under Lakatos at the London School of Economics, developing the latter’s methodology of research programs and testing it against a detailed case history from 19th century physics. Worrall was appointed to a lectureship at the London School of Economics in 1971, becoming professor in 1998.

Shing-Tung Yau (1949–). Chinese mathematician who has done highly influential work on differential geometry and partial differential equations. He is an analyst’s geometer (or geometer’s analyst) with enormous technical power and insight. He has cracked problems on which progress has been stopped for years. Yau proved the Calabi conjecture in 1976. Another conjecture proved by Yau was the positive mass conjecture, which comes from Riemannian geometry.

James A. Yorke (1941–). U.S. mathematician whose 1975 paper with T. Y. Li introduced the term “chaos.” This originally referred to iterations which eventually are periodic with all periods. Later it came to be a more general term, encompassing evolutions that have “strange attractors,” and evolutions in which even very small perturbations of their initial conditions can produce very large effects eventually.

Gail S. Young (1915–1999). U.S. mathematician who was a student of Robert Lee Moore at the University of Texas. Young held appointments at Purdue, Michigan, Tulane, Rochester, Wyoming, and Columbia and was chair of mathematics at two of these. He was also a president of the Mathematical Association of America. Moore was famous for his version of student-centered mathematics “instruction,” and Young was one of his many students who became very prominent as researchers and organization officials.

Grace Chisholm Young (1868–1944). English mathematician who, with William Young, wrote 220 mathematical articles and several books. It is almost impossible to tell exactly how much of the work in these papers was due to Grace Young. As William Young wrote himself, “I feel partly as if I were teaching you, and setting you problems which I could not quite do myself but could enable you to.”

William Henry Young (1863–1942). English mathematician who discovered Lebesgue integration, independently but 2 years after Lebesgue. He studied Fourier series and orthogonal series in general.

Oscar Zariski (1899–1986). Ukrainian-American professor of mathematics at Harvard who worked on the foundations of algebraic geometry using algebraic methods. He contributed to the theory of normal varieties, local uniformization, and the reduction of singularities of algebraic varieties.

Doron Zeilberger (1950–). Israeli-American professor of mathematics at Rutgers Unniversity who has made numerous important contributions to combinatorics, hypergeometric identities, and q-series. Together with Herbert Wilf, Zeilberger was awarded the AMS Steele Prize for their development of WZ theory, which has revolutionized the field of hypergeometric summation. Zeilberger is known as a champion of using computers and algorithms to do mathematics quickly and efficiently. He credits his computer “Shalosh B. Ekhad” as a coauthor. (“Shalosh” and “Ekhad” mean “Three” and “One” in Hebrew respectively, referring to the AT&T 3B1 model).

Andrei Zelevinsky. Russian-American professor of mathematics at Northeastern University, researcher in algebra and combinatorics, and instructor at the Jewish People’s University.