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, mathematical structure, structure, space, Mathematica ...

, best known for two

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

s. He was born in Waltdorf (near Neiße,

Upper Silesia Upper Silesia ( ; ; ; ; Silesian German: ; ) is the southeastern part of the historical and geographical region of Silesia, located today mostly in Poland, with small parts in the Czech Republic. The area is predominantly known for its heav ...

, now

Nysa, Poland Nysa ( or ''Neiße'') is a city in southern 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 comprises the urban portion of ...

), and died in Sulzburg-Laufen,

West Germany West Germany was the common English name for the Federal Republic of Germany (FRG) from its formation on 23 May 1949 until German reunification, its reunification with East Germany on 3 October 1990. It is sometimes known as the Bonn Republi ...

. He was a student at Carolinum in Nysa and then

Hamburg University The University of Hamburg (, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vorlesungswesen''), the Hamburg Colon ...

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 (; ; ; ; ; ; , ) is the historic Germany, German and Prussian name of the city now called Kaliningrad, Russia. The city was founded in 1255 on the site of the small Old Prussians, Old Prussian settlement ''Twangste'' by the Teuton ...

in 1934, and subsequently held posts in a number of universities until 1974.

Sperner's theorem Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory. It is named after Emanuel Sperner, wh ...

, 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 ...

) 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 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 m ...

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 o ...

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 coordi ...

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

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 Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the ...

without explicit use of homology. Sperner's students included Kurt Leichtweiss and Gerhard Ringel.

References

External links

Sperner's photos
– from the

Mathematical Research Institute of Oberwolfach The Oberwolfach Research Institute for Mathematics () is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekly workshops on diverse topics where mathematicians and ...

{{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 Academic staff of the University of Freiburg University of Hamburg alumni Academic staff of the University of Hamburg Academic staff of the University of Königsberg Academic staff of the University of Strasbourg Academic staff of the University of Bonn Presidents of the German Mathematical Society