Emanuel Sperner (9 December 1905 – 31 January 1980) was a German

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, best known for two

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

s. He was born in Waltdorf (near Neiße,

Upper Silesia Upper Silesia ( pl, Górny Śląsk; szl, Gůrny Ślůnsk, Gōrny Ślōnsk; cs, Horní Slezsko; german: Oberschlesien; Silesian German: ; la, Silesia Superior) is the southeastern part of the historical and geographical region of Silesia, located ...

, now

Nysa, Poland Nysa (german: Neisse or ''Neiße'', szl, Nysa) is a town in southwestern Poland on the Eastern Neisse ( Polish: ''Nysa Kłodzka'') river, situated in the Opole Voivodeship. With 43,849 inhabitants (2019), it is the capital of Nysa County. It ...

), and died in Sulzburg-Laufen,

West Germany West Germany is the colloquial term used to indicate the Federal Republic of Germany (FRG; german: Bundesrepublik Deutschland , BRD) between its formation on 23 May 1949 and the German reunification through the accession of East Germany on 3 O ...

. He was a student at Carolinum in Nysa and then Hamburg University where his advisor was

Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...

. He was appointed Professor in

Königsberg Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was named ...

in 1934, and subsequently held posts in a number of universities until 1974. Sperner's theorem, from 1928, says that the size of an

antichain In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its wid ...

in the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

of an ''n''-set (a

Sperner family In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family ''F'' of subsets of a finite set ''E'' in which none of the sets contains another. Equivalently, a Sperner family is an antichain i ...

) is at most the middle

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

(s). It has several proofs and numerous generalizations, including the Sperner property of a partially ordered set.

Sperner's lemma In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an ...

, from 1928, states that every

Sperner coloring In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an ...

of a

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

of an ''n''-dimensional

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

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 Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...

characterizing

dimensionality In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

s. It was later noticed that this lemma provides a direct proof of the

Brouwer fixed-point theorem 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 f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...

without explicit use of

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

. Sperner's students included

Kurt Leichtweiss Kurt Leichtweiß (March 2, 1927 in Villingen-Schwenningen – June 23, 2013) was a mathematician specializing in convex and differential geometry. In 1944, while still in high school Leichtweiß traveled to the Oberwolfach Research Institute for ...

and

Gerhard Ringel Gerhard Ringel (October 28, 1919 in Kollnbrunn, Austria – June 24, 2008 in Santa Cruz, California) was a German mathematician. He was one of the pioneers in graph theory and contributed significantly to the proof of the Heawood conjecture (now ...

References

External links

Sperner's photos
– from the

Mathematical Research Institute of Oberwolfach The Oberwolfach Research Institute for Mathematics (german: Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekl ...

{{DEFAULTSORT:Sperner, Emanuel 1905 births 1980 deaths People from Nysa County People from the Province of Silesia 20th-century German mathematicians Combinatorialists Kolegium Carolinum Neisse alumni University of Freiburg alumni University of Freiburg faculty University of Hamburg alumni University of Hamburg faculty University of Königsberg faculty University of Strasbourg faculty University of Bonn faculty