Karl Menger

Last updated
Karl Menger
Karl Menger 1970 Shimer College Wiki.jpg
Born(1902-01-13)January 13, 1902
DiedOctober 5, 1985(1985-10-05) (aged 83)
Nationality Austrian
Alma mater University of Vienna
Known for Menger sponge
Menger's theorem
Distance geometry
Scientific career
Fields Mathematics
Institutions Illinois Institute of Technology
University of Notre Dame
University of Vienna
Thesis Über die Dimensionalität von Punktmengen (1924)
Doctoral advisor Hans Hahn
Doctoral students Abraham Wald
Witold Hurewicz

Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician. He was the son of the economist Carl Menger. He is credited with Menger's theorem. He worked on mathematics of algebras, algebra of geometries, curve and dimension theory, etc. Moreover, he contributed to game theory and social sciences.

Contents

Biography

Karl Menger was a student of Hans Hahn and received his PhD from the University of Vienna in 1924. L. E. J. Brouwer invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and The Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he was a professor at Illinois Institute of Technology in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree. [1]

Contributions to mathematics

Computer illustration of the "Menger sponge". Menger sponge (IFS).jpg
Computer illustration of the "Menger sponge".

His most famous popular contribution was the Menger sponge (mistakenly known as Sierpinski's sponge), a three-dimensional version of Sierpinski's carpet. It is also related to the Cantor set.

With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values. The characteristic mathematical expressions appearing in those definitions are Cayley–Menger determinants.

He was an active participant of the Vienna Circle which had discussions in the 1920s on social science and philosophy. During that time, he published an influential result [2] on the St. Petersburg paradox with applications to the utility theory in economics; this result has since been criticised as fundamentally misleading. [3] Later he contributed to the development of game theory with Oskar Morgenstern.

Legacy

Menger's longest and last academic post was at the Illinois Institute of Technology, which hosts an annual IIT Karl Menger Lecture and offers the IIT Karl Menger Student Award to an exceptional student for scholarship each year. [4]

See also

Notes

  1. "Biography of Karl Menger". Illinois Institute of Technology . Retrieved 2010-12-22.
  2. Menger, Karl (1934-08-01). "Das Unsicherheitsmoment in der Wertlehre". Zeitschrift für Nationalökonomie (in German). 5 (4): 459–485. doi:10.1007/BF01311578. ISSN   1617-7134. S2CID   151290589.
  3. Peters, O. and Gell-Mann, M., 2016. Evaluating gambles using dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(2), p.023103
  4. "Remembering Karl Menger". Illinois Institute of Technology. Archived from the original on 2009-04-02. Retrieved 2009-03-26.

Further reading

Related Research Articles

Carl Menger Founder of the Austrian School of economics (1840–1921)

Carl Menger von Wolfensgrün was an Austrian economist and the founder of the Austrian School of economics. Menger contributed to the development of the theories of marginalism and marginal utility, which rejected cost-of-production theory of value, such as developed by the classical economists such as Adam Smith and David Ricardo. As a departure from such, he would go on to call his resultant perspective, the subjective theory of value.

Differential geometry Branch of mathematics dealing with functions and geometric structures on differentiable manifolds

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity, as it relates to astronomy and the geodesy of the Earth, and later in the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th century and the 19th century.

Sierpiński carpet Plane fractal built from squares

The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust.

Topology Branch of mathematics that deals with continuous deformations

In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

Alfred Tarski American mathematician

Alfred Tarski was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.

Menger sponge

In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

Shing-Tung Yau Chinese mathematician

Shing-Tung Yau is an American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.

Arthur Cayley English mathematician (1821–1895)

Arthur Cayley was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.

Witold Hurewicz was a Polish mathematician.

Distance geometry is the characterization and study of sets of points based only on given values of the distances between member pairs. More abstractly, it is the study of semimetric spaces and the isometric transformations between them. In this view, it can be considered as a subject within general topology.

Karl Georg Christian von Staudt German mathematician

Karl Georg Christian von Staudt was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.

Chennai Mathematical Institute

Chennai Mathematical Institute (CMI) is a research and education institute in Chennai, India. It was founded by the SPIC Science Foundation in 1989, and offers undergraduate as well as postgraduate programmes in physics, mathematics and computer science, besides its key strength in the form of high-end research in Mathematics. CMI is noted for its research in the field of algebraic geometry, in particular in the area of moduli of bundles.

Menger is a surname. Notable people with the surname include:

Franz Leopold Alt was an Austrian-born American mathematician who made major contributions to computer science in its early days. He was best known as one of the founders of the Association for Computing Machinery, and served as its president from 1950 to 1952.

David Eisenbud American mathematician

David Eisenbud is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and was Director of the Mathematical Sciences Research Institute (MSRI) from 1997 to 2007. He was reappointed to this office in 2013, and his term has been extended until July 31, 2022.

Margaret Wertheim

Margaret Wertheim is an Australian-born science writer, curator, and artist based in the United States. She is the author of books on the cultural history of physics, and has written about science, including for the New York Times, Los Angeles Times, Guardian, Aeon and Cabinet. Wertheim and her twin sister, Christine Wertheim, are co-founders of the Institute For Figuring (IFF), a Los Angeles-based non-profit organization though which they create projects at the intersection of art, science and mathematics. Their IFF projects include their Crochet Coral Reef, which has been shown at the 2019 Venice Biennale, Hayward Gallery (London), Museum of Arts and Design (NYC), and the Smithsonian's National Museum of Natural History. For her work with public science engagement, Wertheim won the 2016 Klopsteg Memorial Award from the American Association of Physics Teachers and Australia's Scientia Medal (2017).

Herbert Federer was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.

Georg Nöbeling German mathematician

Georg August Nöbeling was a German mathematician.

Geometry Branch of mathematics

Geometry is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.

Trivandrum Ramakrishnan "T. R." Ramadas is an Indian mathematician who specializes in algebraic and differential geometry, and mathematical physics. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in 1998, the highest science award in India, in the mathematical sciences category.