Emanuel Sperner
Encyclopedia
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, best known for two theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s. He was born in Waltdorf (near Neiße, Upper Silesia
Upper Silesia
Upper Silesia is the southeastern part of the historical and geographical region of Silesia. Since the 9th century, Upper Silesia has been part of Greater Moravia, the Duchy of Bohemia, the Piast Kingdom of Poland, again of the Lands of the Bohemian Crown and the Holy Roman Empire, as well as of...
, now Nysa, Poland
Nysa, Poland
Nysa is a town in southwestern Poland on the Nysa Kłodzka river with 47,545 inhabitants , situated in the Opole Voivodeship. It is the capital of Nysa County. It comprises the urban portion of the surrounding Gmina Nysa, a mixed urban-rural commune with a total population of 60,123 inhabitants...
), and died in Sulzburg-Laufen
Sulzburg
Sulzburg is a town in the district Breisgau-Hochschwarzwald, in Baden-Württemberg, Germany. It is situated on the western slope of the Black Forest, 20 km southwest of Freiburg.Sulzburg's lovely, barrel-vaulted synagogue has been completely restored....
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
. He was a student at Hamburg University where his advisor was Wilhelm Blaschke
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
. He was appointed Professor in Königsberg
Königsberg
Königsberg was the capital of East Prussia from the Late Middle Ages until 1945 as well as the northernmost and easternmost German city with 286,666 inhabitants . Due to the multicultural society in and around the city, there are several local names for it...
in 1934, and subsequently held posts in a number of universities until 1974.
The Sperner theorem
Sperner family
In combinatorics, a Sperner family , named in honor of Emanuel Sperner, is a set system in which no element is contained in another. Formally,...
, from 1928, says that the size of an antichain
Antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. Let S be a partially ordered set...
in the power set of an n-set is at most the middle binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
(s). It has several proofs and numerous generalizations.
Sperner's lemma
Sperner's lemma
In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors...
, from 1928, states that every Sperner coloring of a triangulation of an n-dimensional simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
contains a cell colored with a complete set of colors. It was proven by Sperner to provide an alternate proof of a theorem of Lebesgue characterizing dimensionality of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s. It was later noticed that this lemma provides a direct proof of the Brouwer fixed-point theorem without explicit use of homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
.
Sperner's students included Kurt Leichtweiss
Kurt Leichtweiss
Kurt Leichtweiss is a mathematician specializing in convex and differential geometry.Leichtweiss was born in Villingen-Schwenningen. In 1944, while still in high school he traveled to the Mathematical Institute where his mathematical interests were encouraged. He studied at the University of...
and Gerhard Ringel
Gerhard Ringel
Gerhard Ringel was a German mathematician who earned his Ph.D. from the University of Bonn in 1951...
.
External links
- Sperner's photos – from the Mathematical Research Institute of OberwolfachMathematical Research Institute of OberwolfachThe Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany, was founded by mathematician Wilhelm Süss in 1944...