**Hans Freudenthal** (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education.^{ [1] }

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.^{ [2] }^{ [3] } He met Brouwer in 1927, when Brouwer came to Berlin to give a lecture, and in the same year Freudenthal also visited the University of Paris.^{ [3] }^{ [4] } He completed his thesis work with Heinz Hopf at Berlin, defended a thesis on the ends of topological groups in 1930, and was officially awarded a degree in October 1931.^{ [2] }^{ [3] }^{ [5] } After defending his thesis in 1930, he moved to Amsterdam to take up a position as assistant to Brouwer.^{ [2] }^{ [3] } In this pre-war period in Amsterdam, he was promoted to lecturer at the University of Amsterdam,^{ [3] }^{ [4] } and married his wife, Suus Lutter, a Dutch teacher.^{ [2] }

Although he was a German Jew, Freudenthal's position in the Netherlands insulated him from the anti-Jewish laws that had been passed in Germany beginning with the Nazi rise to power in 1933.^{ [3] } However, in 1940 the Germans invaded the Netherlands, following which Freudenthal was suspended from duties at the University of Amsterdam by the Nazis.^{ [3] }^{ [4] } 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 a non-Jew, had not been deported) he escaped in 1944 and went into hiding with his family in occupied Amsterdam.^{ [6] } During this period Freudenthal occupied his time in literary pursuits, including winning first prize under a false name in a novel-writing contest.^{ [3] }

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 the rest of his career.^{ [2] }^{ [3] } He served as the 8th president of the International Commission on Mathematical Instruction from 1967 to 1970.^{ [7] } 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.^{ [3] } In 1972 he founded and became editor-in-chief of the journal * Geometriae Dedicata *.^{ [8] } He retired from his professorship in 1975^{ [3] } and from his journal editorship in 1981.^{ [8] } He died in Utrecht in 1990, sitting on a bench in a park where he always took a morning walk.^{ [2] }

In his thesis work, published as a journal article in 1931, Freudenthal introduced the concept of an end of a topological space.^{ [9] } 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,^{ [10] } and also in other areas of mathematics such as 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.^{ [11] } 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 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.^{ [12] } 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.^{ [13] }

In 1968, Freudenthal founded 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.^{ [14] }

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".^{ [2] } He was also a fervent critic of one of the first international school achievement studies.^{ [15] } 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).^{ [16] }

Freudenthal published the Impossible Puzzle, a mathematical puzzle that appears to lack sufficient information for a solution, in 1969.^{ [17] } He also designed a constructed language, Lincos, to make possible communication with extraterrestrial intelligence.^{ [18] }^{ [19] }

- Freudenthal, Hans (1931), "Über die Enden topologischer Räume und Gruppen",
*Mathematische Zeitschrift*,**33**: 692–713, doi:10.1007/BF01174375, S2CID 120965216, Zbl 0002.05603 - Freudenthal, Hans (1936), "Teilweise geordnete Moduln" (PDF),
*Proc. Akad. Wet. Amsterdam*(in German),**39**: 641–651, Zbl 0014.31302 . - Freudenthal, Hans (1937), "Über die Klassen der Sphärenabbildungen. I. Große Dimensionen",
*Compositio Mathematica*(in German),**5**: 299–314, Zbl 0018.17705 . - Freudenthal, Hans (1960),
*Lincos: design of a language for cosmic intercourse*, Studies in logic and the foundations of mathematics, North-Holland Pub. Co.. - Freudenthal, Hans; de Vries, H. (1969),
*Linear Lie Groups*, Pure and Applied Mathematics,**35**, New York: Academic Press, MR 0260926 . - Freudenthal, Hans (1972),
*Mathematics as an Educational Task*, Springer, ISBN 9789027702357 . - Freudenthal, Hans (1986),
*Didactical Phenomenology of Mathematical Structures*, Mathematics Education Library,**1**, Springer, ISBN 9789027722614 . - Freudenthal, Hans (1991),
*Revisiting Mathematics Education: China Lectures*, Kluwer Academic Publishers, ISBN 0-7923-1299-6 . - Freudenthal, Hans; Freudenthal, Matías (2015),
*El viaje de Ofantito*(in Spanish), Granada (Spain): Esdrújula Ediciones, ISBN 978-84-164850-9-3 [Children's story left unfinished in 1943, completed and translated into Spanish by his son Matijs (Matías)].^{ [20] }

In 1951, Freudenthal became a member of the Royal Netherlands Academy of Arts and Sciences.^{ [21] } He was also an honorary member of the International Academy of the History of Science.^{ [4] } He was awarded the Gouden Ganzenveer award in 1984.^{ [22] }

In 2000, the International Commission on Mathematical Instruction instituted an award named in honor of Freudenthal, 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.^{ [7] }

The asteroid 9689 Freudenthal is named after him.^{ [23] }

**Brouwer's fixed-point theorem** is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a convex compact subset of Euclidean space to itself.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

**Luitzen Egbertus Jan Brouwer**, usually cited as **L. E. J. Brouwer** but known to his friends as **Bertus**, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis. He is known as the founder of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension.

In differential geometry, the **Atiyah–Singer index theorem**, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the **analytical index** is equal to the **topological index**. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

**Witold Hurewicz** was a Polish mathematician.

**Pierre René, Viscount Deligne** is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.

**Wilhelm Wirtinger** was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.

In mathematics, the **principal ideal theorem** of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called *principalization*, or sometimes *capitulation*.

**Phillip Augustus Griffiths IV** is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory. He also worked on partial differential equations, coauthored with Shiing-Shen Chern, Robert Bryant and Robert Gardner on Exterior Differential Systems.

**Johannes de Groot** was a Dutch mathematician, the leading Dutch topologist for more than two decades following World War II.

**John Robert Stallings Jr.** was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.

**Michael Jerome Hopkins** is an American mathematician known for work in algebraic topology.

* Geometriae Dedicata* is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the Netherlands. It is published by Springer Netherlands. The Editors-in-Chief are John R. Parker and Jean-Marc Schlenker.

In mathematics, **Hurwitz's theorem** is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called **Hurwitz algebras**, are examples of composition algebras.

In algebra, **Freudenthal algebras** are certain Jordan algebras constructed from composition algebras.

**Lawrence Man Hou Ein** is a mathematician who works in algebraic geometry.

In mathematics, **nonabelian algebraic topology** studies an aspect of algebraic topology that involves higher-dimensional algebras.

**Ascher Otto Wagner** was an Austrian and British mathematician, specializing in the theory of finite groups and finite projective planes. He is known for the Dembowski–Wagner theorem.

In mathematics, particularly algebraic topology, the **Kan-Thurston theorem** associates a discrete group to every path connected topological space in such a way that the group cohomology of is the same as the cohomology of the space . The group might then be regarded as a good approximation to the space , and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory.

**Guido Mislin** is a Swiss mathematician, academic and researcher. He is a Professor Emeritus of Mathematics at ETH Zurich. He is also associated with Ohio State University as a guest at Mathematics Department.

- ↑ Streefland, Leen, ed. (1994),
*The Legacy of Hans Freudenthal*, Springer, ISBN 9780792326533 . - 1 2 3 4 5 6 7
*Prof. Dr. Hans Freudenthal (1905 - 1990)*, Freudenthal institute for science and mathematics education, archived from the original on 2013-10-20, retrieved 2013-01-10. - 1 2 3 4 5 6 7 8 9 10 11 O'Connor, John J.; Robertson, Edmund F., "Hans Freudenthal",
*MacTutor History of Mathematics archive*, University of St Andrews - 1 2 3 4 Bos, Henk J. M. (1992), "In memoriam Hans Freudenthal (1905–1990)",
*Historia Mathematica*,**19**(1): 106–108, doi: 10.1016/0315-0860(92)90070-R . - ↑ Hans Freudenthal at the Mathematics Genealogy Project
- ↑ Müller-Hoissen, Folkert; Pallo, Jean Marcel (2012),
*Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift*, Progress in Mathematics,**299**, Springer, p. 20, ISBN 9783034804059 . - 1 2
*The ICMI Felix Klein and Hans Freudenthal Awards*, International Commission on Mathematical Instruction, archived from the original on 2013-01-27, retrieved 2013-01-10. - 1 2 Veldkamp, F. D. (1985), "In honor of Hans Freudenthal on his eightieth birthday",
*Geometriae Dedicata*,**19**(1): 3–5, doi:10.1007/BF00233100, S2CID 122234088 . - ↑ Streefland (1994), pp. 60–61.
- ↑ James, I. M. (1999),
*History of Topology*, Elsevier, p. 1004, ISBN 9780080534077 . - ↑ Lavrič, B. (1986), "On Freudenthal's spectral theorem",
*Koninklijke Nederlandse Akademie van Wetenschappen*,**48**(4): 411–421, MR 0869757 . - ↑ Whitehead, G. W. (1953), "On the Freudenthal Theorems",
*Annals of Mathematics*,**57**(2): 209–228, doi:10.2307/1969855, JSTOR 1969855, MR 0055683 . - ↑ Baez, John C. (2002), "The Octonions",
*Bulletin of the American Mathematical Society*,**39**(2): 145–205, arXiv: math/0105155v4 , doi:10.1090/S0273-0979-01-00934-X, MR 1886087, S2CID 586512 . - ↑ Beckers, Danny (2019-02-05). "Why to publish on mathematics education so as to be useful? Educational Studies in Mathematics and its founder Hans Freudenthal".
*Educational Studies in Mathematics*.**101**: 7–17. doi: 10.1007/s10649-019-9881-4 . ISSN 1573-0816. - ↑
*Pupils achievements internationally compared — the IEA.*Educational Studies in Mathematics 6, 127–186. According to PISA critic Joachim Wuttke, some of the points Freudenthal made still apply to the most recent international school studies: unequal enrollment rates, the unsolved translation problem, lacking curricular validity, reading items that contain deeper science than the science items, overinterpretation of numerical outcomes, Kafkaesk confusion in the documentation and in the underlying decisions, dogmatic rejection of criticism . - ↑ Freudenthal, Hans (1972),
*Mathematics as an Educational Task*, Springer, ISBN 9789027702357 - ↑ Gardner, Martin (December 1979), "Mathematical Games: A Pride of Problems, Including One That Is Virtually Impossible",
*Scientific American*,**241**: 22–30, doi:10.1038/scientificamerican0979-22 . - ↑ DeVito, Carl L. (1992), "Languages, Science and the Search for Extraterrestrial Intelligence",
*Leonardo*,**25**(1): 13–16, doi:10.2307/1575613, JSTOR 1575613 . - ↑ Narens, Louis; Freudenthal, Hans (1973), "Review: Hans Freudenthal, Lincos. Design of a Language for Cosmic Intercourse. Part I",
*Journal of Symbolic Logic*,**38**(3): 517, doi:10.2307/2273050, JSTOR 2273050 . - ↑ Esdrújula (2015) Publisher file of
*El viaje de Ofantito*(in Spanish) retrieved 18 September 2015. - ↑ "Hans Freudenthal (1905 - 1990)". Royal Netherlands Academy of Arts and Sciences. Retrieved 27 July 2015.
- ↑ Stichting De Gouden Ganzenveer, retrieved 2013-01-10.
- ↑ JPL Small-Body Database Browser on 9689 Freudenthal

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- Autobiography by Hans Freudenthal, Hans, schrijf dat op
- Strambach, Karl; Veldkamp, Ferdinand D. (1981), "A tribute to Hans Freudenthal",
*Geometriae Dedicata*,**11**(1): 125, doi:10.1007/BF00183195, S2CID 121711425 . - Howson, G. (1985), "Hans Freudenthal and the foundations of a discipline of mathematics education",
*Nieuwe Wiskrant*,**5**: 68–72. - Van Est, W. T. (1993), "Hans Freudenthal (17 September 1905–13 October 1990)",
*Educational Studies in Mathematics*,**25**(1–2): 59–69, doi:10.1007/BF01274102, S2CID 144711501 . - Strambach, Karl; Veldkamp, Ferdinand D. (1991), "In memoriam Hans Freudenthal",
*Geometriae Dedicata*,**37**(2): 119, doi:10.1007/BF00147407, S2CID 122967603 . - Gravemeijer, K.; Terwel, J. (2000), "Hans Freudenthal: A mathematician on didactics and curriculum theory",
*Journal of Curriculum Studies*,**32**(6): 777–796, doi:10.1080/00220270050167170, S2CID 55632373 .

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