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26-569: Freudenthal was born in Luckenwalde , Brandenburg , on 17 September 1905, the son of a Jewish teacher. He was interested in both mathematics and literature as a child, and studied mathematics at the University of Berlin beginning in 1923. He met L. E. J. Brouwer in 1927, when Brouwer came to Berlin to give a lecture, and in the same year Freudenthal also visited the University of Paris . He completed his thesis work with Heinz Hopf at Berlin, defended

52-555: A bench in a park where he always took a morning walk. In his thesis work, published as a journal article in 1931, Freudenthal introduced the concept of an end of a topological space . Ends are intended to capture the intuitive idea of a direction in which the space extends to infinity, but have a precise mathematical formulation in terms of covers of the space by nested sequences of compact sets . Ends remain of great importance in topological group theory, Freudenthal's motivating application, and also in other areas of mathematics such as

78-521: A foreign member of the Royal Netherlands Academy of Arts and Sciences in 1988. He introduced the theory of buildings (sometimes known as Tits buildings ), which are combinatorial structures on which groups act, particularly in algebraic group theory (including finite groups , and groups defined over the p-adic numbers ). The related theory of (B, N) pairs is a basic tool in the theory of groups of Lie type . Of particular importance

104-468: A key manufacturer of hats. In 1921 the two biggest hat ateliers, Herrmann and Steinberg, merged and set up their factory on an industrial estate in Luckenwalde. The factory was designed by German architect Erich Mendelsohn in 1923, the factory is considered a milestone of Expressionist architecture . The hat factory fell into disrepair during and after the war period and was restored in 2001, but as of 2013

130-554: A non-Jew, had not been deported) he escaped in 1944 and went into hiding with his family in occupied Amsterdam. During this period Freudenthal occupied his time in literary pursuits, including winning first prize under a false name in a novel-writing contest. With the war over, Freudenthal's position at the University of Amsterdam was returned to him, but in 1946 he was given a chair in pure and applied mathematics and foundations of mathematics at Utrecht University , where he remained for

156-742: A solution, in 1969. He also designed a constructed language , Lincos , to make possible communication with extraterrestrial intelligence . In 1951, Freudenthal became a member of the Royal Netherlands Academy of Arts and Sciences . He was also an honorary member of the International Academy of the History of Science . He was awarded the Gouden Ganzenveer award in 1984. In 2000, the International Commission on Mathematical Instruction instituted an award named in honor of Freudenthal,

182-443: A suitable group of symmetries (the so-called Moufang polygons ). In collaboration with François Bruhat he developed the theory of affine buildings, and later he classified all irreducible buildings of affine type and rank at least four. Another of his well-known theorems is the " Tits alternative ": if G is a finitely generated subgroup of a linear group , then either G has a solvable subgroup of finite index or it has

208-524: A thesis on the ends of topological groups in 1930, and was officially awarded a degree in October 1931. After defending his thesis in 1930, he moved to Amsterdam to take up a position as assistant to Brouwer. In this pre-war period in Amsterdam, he was promoted to lecturer at the University of Amsterdam , and married his wife, Suus Lutter, a Dutch teacher. Although he was a German Jew, Freudenthal's position in

234-532: Is his classification of all irreducible buildings of spherical type and rank at least three, which involved classifying all polar spaces of rank at least three. The existence of these buildings initially depended on the existence of a group of Lie type in each case, but in joint work with Mark Ronan he constructed those of rank at least four independently, yielding the groups directly. In the rank-2 case spherical building are generalized n-gons , and in joint work with Richard Weiss he classified these when they admit

260-701: Is the capital of the Teltow-Fläming district in the German state of Brandenburg . It is situated on the Nuthe river north of the Fläming Heath , at the eastern rim of the Nuthe-Nieplitz Nature Park , about 50 km (31 mi) south of Berlin . The town area includes the villages of Frankenfelde and Kolzenburg. The former Slavic settlement of Lugkin was conquered by Margrave Conrad Wettin of Meissen in

286-737: The Deutsche Mathematiker-Vereinigung (German Mathematical Society) in 1996, and the German distinction " Pour le Mérite ". In 2008 he was awarded the Abel Prize , along with John Griggs Thompson , "for their profound achievements in algebra and in particular for shaping modern group theory". Tits became a member of the French Academy of Sciences in 1979. He was a member of the Norwegian Academy of Science and Letters . He became

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312-696: The Tits metric . Tits was born in Uccle , Belgium to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels . His thesis advisor was Paul Libois , and Tits graduated with his doctorate in 1950 with the dissertation Généralisation des groupes projectifs basés sur la notion de transitivité . Tits held professorships at the Free University of Brussels (now split into

338-678: The Université Libre de Bruxelles and the Vrije Universiteit Brussel ) (1962–1964), the University of Bonn (1964–1974) and the Collège de France in Paris , until becoming emeritus in 2000. He changed his citizenship to French in 1974 in order to teach at the Collège de France, which at that point required French citizenship. Because Belgian nationality law did not allow dual nationality at

364-480: The homotopy groups of spheres (since every sphere can be formed topologically as a suspension of a lower-dimensional sphere) and eventually formed the basis of stable homotopy theory . The Freudenthal magic square is a construction in Lie algebra developed by Freudenthal (and independently by Jacques Tits ) in the 1950s and 1960s, associating each Lie algebra to a pair of division algebras . In 1968, Freudenthal founded

390-414: The industries Luckenwalde was known for in the early 20th century. http://webgerman.com/Notgeld/Directory/L/Luckenwalde.htm Jacques Tits Jacques Tits ( French: [ʒak tits] ) (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry . He introduced Tits buildings , the Tits alternative , the Tits group , and

416-464: The 17th century, Luckenwalde's cloth and wool factories did not spring up till the reign of King Frederick II of Prussia and soon were among the most extensive in Germany. Other traditional industries were cotton printing and a dye works, brewing, and the making of metal and bronze goods. In 1808 Luckenwalde officially received town privileges. By the turn of the 20th Century Luckenwalde became renowned as

442-676: The Hans Freudenthal Medal. It is given in odd-numbered years (beginning in 2003) for an "outstanding achievement in mathematics education research" in the form of "a major cumulative program of research". Recipients of the medal have included Celia Hoyles , Paul Cobb, Anna Sfard , Yves Chevallard , Luis Radford and Frederick Leung . The asteroid 9689 Freudenthal is named after him. Luckenwalde Luckenwalde ( German: [ˌlʊkn̩ˈvaldə] ; Upper and Lower Sorbian : Łukowc , Upper Sorbian: [ˈwukɔfts] , Lower Sorbian: [ˈwukɔwts] )

468-625: The Netherlands insulated him from the anti-Jewish laws that had been passed in Germany beginning with the Nazi rise to power in 1933. However, in 1940 the Germans invaded the Netherlands , following which Freudenthal was suspended from duties at the University of Amsterdam by the Nazis. In 1943 Freudenthal was sent to a labor camp in the village of Havelte in the Netherlands, but with the help of his wife (who, as

494-401: The building remains empty. During World War II , there was a Stalag for prisoners of war ( Stalag III-A ). There was also a work camp for civilians. The Nazis forced people to work for their war effort or else the families of people who worked there would perish. Lack of food and hard work killed thousands. Among them were Poles, Italians, French and many more. There were several places in

520-626: The course of the 1147 Wendish Crusade . Lukenwalde Castle was first mentioned in a 1216 deed as a burgward of the Bishopric of Brandenburg , it was acquired by Zinna Abbey in 1285. Together with Zinna it remained under the rule of the Archbishopric of Magdeburg and its successor, the Prussian Duchy of Magdeburg until it was attached to the Margraviate of Brandenburg in 1773. Originating in

546-484: The journal, Educational Studies in Mathematics (ESM). Becoming one of the top-rated journals in the field of mathematics education, ESM was focused on publishing research around finding better ways to teach mathematics. Later in his life, Freudenthal focused on elementary mathematics education . In the 1970s, his single-handed intervention prevented the Netherlands from following the worldwide trend of " new math ". He

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572-735: The rest of his career. He served as the 8th president of the International Commission on Mathematical Instruction from 1967 to 1970. In 1971 he founded the Institute for the Development of Mathematical Education (IOWO) at Utrecht University, which after his death was renamed the Freudenthal Institute . In 1972 he founded and became editor-in-chief of the journal Geometriae Dedicata . He retired from his professorship in 1975 and from his journal editorship in 1981. He died in Utrecht in 1990, sitting on

598-525: The study of minimal surfaces . In 1936, while working with Brouwer, Freudenthal proved the Freudenthal spectral theorem on the existence of uniform approximations by simple functions in Riesz spaces . In 1937 he proved the Freudenthal suspension theorem , showing that the suspension operation on topological spaces shifts by one their low-dimensional homotopy groups ; this result was important in understanding

624-586: The time, he renounced his Belgian citizenship. Tits was an "honorary" member of the Nicolas Bourbaki group; as such, he helped popularize H.S.M. Coxeter 's work, introducing terms such as Coxeter number , Coxeter group , and Coxeter graph . Tits died on 5 December 2021, at the age of 91 in the 13th arrondissement , Paris. Tits received the Wolf Prize in Mathematics in 1993, the Cantor Medal from

650-780: The town and surrounding areas where they worked. Luckenwalde was taken by the Red Army on 22 April 1945. Seats in the municipal assembly ( Stadtverordnetenversammlung ) as of 2014 elections: Luckenwalde station is located on the Berlin–Halle railway . Luckenwalde is twinned with: [REDACTED]   This article incorporates text from a publication now in the public domain :  Chisholm, Hugh , ed. (1911). " Luckenwalde ". Encyclopædia Britannica . Vol. 17 (11th ed.). Cambridge University Press. p. 106. [REDACTED] Media related to Luckenwalde at Wikimedia Commons Notgeld (emergency banknotes) depicting

676-526: Was also a fervent critic of one of the first international school achievement studies. He interpreted mathematics as a human activity where students should open a scientific eye on the world around them, mathematizing real situations, in a context that makes sense for the students. This approach is called Realistic Mathematics Education (RME). Freudenthal published the Impossible Puzzle , a mathematical puzzle that appears to lack sufficient information for

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