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Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. Life He was born in Moscow, the son of Alexander Lefschetz and his wife Sarah or Vera Lifschitz, Jewish traders who used to travel around Europe and the Middle East (they held Ottoman passports). Shortly thereafter, the family moved to Paris. He was educated there in engineering at the École Centrale Paris, but emigrated to the US in 1905. He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas, moving to Princeton University in 1924, where he was soon given a permanent position. He rema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Fixed-point Theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926. The counting is subject to an imputed multiplicity at a fixed point called the fixed-point index. A weak version of the theorem is enough to show that a mapping without ''any'' fixed point must have rather special topological properties (like a rotation of a circle). Formal statement For a formal statement of the theorem, let :f\colon X \rightarrow X\, be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number \Lambda_f of f by :\Lambda_f:=\sum_(-1)^k\mathrm(f_*, H_k(X,\Q)), the alternating (finite) sum of the matrix traces of the linear maps induced by f on H_k(X,\Q), the singular homology groups of X with rational coefficients. A simple vers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard–Lefschetz Theory
In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical point (mathematics), critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces in his book , and extended to higher dimensions by . It is a complex analog of Morse theory that studies the topology of a real manifold by looking at the critical points of a real function. extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the Weil conjectures. Picard–Lefschetz formula The Picard–Lefschetz formula describes the monodromy at a critical point. Suppose that ''f'' is a holomorphic map from an ''(k+1)''-dimensional projective complex manifold to the projective line P1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images ''x''1,...,''x''''n'' in P1. Pick any other point ''x'' in P1. The fundamen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Steenrod
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled ''Universal homology groups''. Steenrod held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He was editor of the Annals of Mathematics and a member of the National Academy of Sciences. He died in Princeton, survived by his wife, the former Carolyn Witter, and two children. Work Thanks to Lefschetz and others, the cup product structure of cohomology was understood by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founded the leading biomathematical journal Mathematical Biosciences. Biography Bellman was born in 1920 in New York City to non-practising Jewish parents of Polish and Russian descent, Pearl (née Saffian) and John James Bellman, who ran a small grocery store on Bergen Street near Prospect Park, Brooklyn. On his religious views, he was an atheist. He attended Abraham Lincoln High School, Brooklyn in 1937,Salvador SanabriaRichard Bellman profile at http://www-math.cudenver.edu retrieved October 3, 2008. and studied mathematics at Brooklyn College where he earned a BA in 1941. He later earned an MA from the University of Wisconsin. During World War II he worked for a Theoretical Physics Division group in Los Alamos. In 1946 he received h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Fox
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played an important role in the modernization and main-streaming of knot theory. Biography Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a PhD degree from Princeton University in 1939. His doctoral dissertation, ''On the Lusternick-Schnirelmann Category'', was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, and supervised Ken ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Edward Story
William Edward Story (April 29, 1850 – April 10, 1930) was an American mathematician who taught at Johns Hopkins University and Clark University. William was born in Boston to Isaac Marion Story (1818-1901) and Elizabeth Bowen Woodberry (1817-1888). He attended high school in Somerville, Massachusetts, and entered Harvard University in the fall of 1867. He graduated with honors in mathematics and began graduate study in Germany in September 1871. In Berlin he attended lectures of Weierstrass, Ernst Kummer, Helmholtz and Dove. In Leipzig he heard Karl Neumann, Bruhns, Mayer, Van der Müll, and Engelmann. He earned a Ph.D. in Leipzig in 1875 with a dissertation "On the algebraic relations existing between the polars of a binary quantic." W.E. Story began his teaching career at Harvard as a tutor. With the establishment of Johns Hopkins University in 1876, Story was recruited by Daniel Coit Gilman as an Associate. J. J. Sylvester led the program in mathematics. Until 1879, St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Begle
Edward Griffith Begle (November 27, 1914 – March 2, 1978) was a mathematician best known for his role as the director of the School Mathematics Study Group (SMSG), the primary group credited for developing what came to be known as The New Math. Begle was a topologist and a researcher in mathematics education who served as a member of the faculty of Stanford University, Princeton University, The University of Michigan, and Yale University. Begle was also elected as the secretary of the American Mathematical Society in 1951, and he held the position for 6 years. Biography Edward G. Begle was born November 27, 1914 in Saginaw, Michigan. Studying at the University of Michigan, Begle earned his A.B. in Mathematics in 1936 and his M.A. in 1938. Begle's early academic work was in the field of topology, which is where he earned his Ph.D. at Princeton, studying under Solomon Lefschetz in 1940. While Begle's contributions to the field of mathematical research are limited, among ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Gomory
Ralph Edward Gomory (born May 7, 1929) is an American applied mathematician and executive. Gomory worked at IBM as a researcher and later as an executive. During that time, his research led to the creation of new areas of applied mathematics. After his career in the corporate world, Gomory became the president of the Alfred P. Sloan Foundation, where he oversaw programs dedicated to broadening public understanding in three key areas: the economic importance of science and research; the effects of globalization on the United States; and the role of technology in education. Gomory has written extensively on the nature of technology development, industrial competitiveness, models of international trade, social issues under current economics and law, and the function of the corporation in a globalizing world. Biography Gomory is the son of Andrew L. Gomory and Marian Schellenberg. He graduated from George School in Newtown, PA in 1946. He received his B.A. from Williams College in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Truesdell
Clifford Ambrose Truesdell III (February 18, 1919 – January 14, 2000) was an American mathematician, natural philosopher, and historian of science. Life Truesdell was born in Los Angeles, California. After high school, he spent two years in Europe learning French, German, and Italian, and improving his Latin and Greek. His linguistic skills stood him in good stead in his later historical investigations. At Caltech he was deeply influenced by the teaching of Harry Bateman. In particular, a course in partial differential equations "taught me the difference between an ordinary good teacher and a great mathematician, and after that I never cared what grade I got in anything." He obtained a B.Sc. in mathematics and physics in 1941, and an MSc. in mathematics in 1942. In 1943, he completed a Ph.D. in mathematics at Princeton University. For the rest of the decade, the U.S. Navy employed him to do mechanics research. Truesdell taught at Indiana University 1950–61, where his students ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Dowker
Clifford Hugh Dowker (; March 2, 1912 – October 14, 1982) was a topologist known for his work in point-set topology and also for his contributions in category theory, sheaf theory and knot theory. Biography Clifford Hugh Dowker grew up on a small farm in Western Ontario, Canada. He excelled in mathematics and was paid to teach his math teacher math at his secondary school. He was awarded a scholarship at Western Ontario University, where he got his B.S. in 1933. He wanted to pursue a career as a teacher, but he was persuaded to continue with his education because of his extraordinary mathematical talent. He earned his M.A. from the University of Toronto in 1936 and his Ph.D. from Princeton University in 1938. His dissertation ''Mapping theorems in non-compact spaces'' was written under the supervision of Solomon Lefschetz and was published (with additions) in 1947 in the '' American Journal of Mathematics''. After earning his doctorate, Dowker became an instructor at the Weste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
