Hellmuth Kneser
Encyclopedia
Hellmuth Kneser was a German
mathematician
, who made notable contributions to group theory
and topology
. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds
. His proof originated the concept of normal surface
, a fundamental cornerstone of the theory of 3-manifold
s.
He was born in Dorpat, Russian Empire
(now Tartu
, Estonia
) and died in Tübingen
, Germany
. He is the son of the mathematician Adolf Kneser
and the father of the mathematician Martin Kneser
. He assisted Wilhelm Süss
in the founding of the Mathematical Research Institute of Oberwolfach
and served as the director of the institute from 1958 to 1959.
Kneser had formulated the problem or non-integer iteration of functions and proved the existence of the entire superfunction
of the exponential
; on the base of this superfunction he constructed the functional square root
of the exponential function as half-iteration of the exponential, i.e. a function φ such that φ(φ(z)) = exp(z).; although this superexponential is not real and cannot be considered as tetration
.
Kneser was a student of David Hilbert
. He was an advisor of a number of notable mathematicians, including Reinhold Baer
.
Hellmuth Kneser was a member of the NSDAP and also the SA
. In July 1934 he wrote to Ludwig Bieberbach
a short note supporting his anti-semitic views and stating: "May God grant German science a unitary, powerful and continued political position."
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....
, who made notable contributions to group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds
Prime decomposition (3-manifold)
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique collection of prime 3-manifolds....
. His proof originated the concept of normal surface
Normal surface
In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad . A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices...
, a fundamental cornerstone of the theory of 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s.
He was born in Dorpat, Russian Empire
Russian Empire
The Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...
(now Tartu
Tartu
Tartu is the second largest city of Estonia. In contrast to Estonia's political and financial capital Tallinn, Tartu is often considered the intellectual and cultural hub, especially since it is home to Estonia's oldest and most renowned university. Situated 186 km southeast of Tallinn, the...
, Estonia
Estonia
Estonia , officially the Republic of Estonia , is a state in the Baltic region of Northern Europe. It is bordered to the north by the Gulf of Finland, to the west by the Baltic Sea, to the south by Latvia , and to the east by Lake Peipsi and the Russian Federation . Across the Baltic Sea lies...
) and died in Tübingen
Tübingen
Tübingen is a traditional university town in central Baden-Württemberg, Germany. It is situated south of the state capital, Stuttgart, on a ridge between the Neckar and Ammer rivers.-Geography:...
, 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 is the son of the mathematician Adolf Kneser
Adolf Kneser
Adolf Kneser was a German mathematician.He was born in Grüssow, Mecklenburg, Germany and died in Breslau, Germany ....
and the father of the mathematician Martin Kneser
Martin Kneser
Martin Kneser was a German mathematician. His father Hellmuth Kneser and grandfather Adolf Kneser were also mathematicians....
. He assisted Wilhelm Süss
Wilhelm Süss
Wilhelm Süss was a German mathematician.He was born in Frankfurt, Germany and died in Freiburg im Breisgau, Germany.He was founder and first director of the Mathematical Research Institute of Oberwolfach.-External links:...
in the founding of the Mathematical Research Institute of Oberwolfach
Mathematical Research Institute of Oberwolfach
The Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany, was founded by mathematician Wilhelm Süss in 1944...
and served as the director of the institute from 1958 to 1959.
Kneser had formulated the problem or non-integer iteration of functions and proved the existence of the entire superfunction
Superfunction
In mathematics, a superfunction is a nonstandard name for an iterated function for complexified continuous iteration index.Roughly, for some function f and for some variable x, the superfunction could be defined by the expression S =...
of the exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
; on the base of this superfunction he constructed the functional square root
Functional square root
In mathematics, a half iterate is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying f = g for all x...
of the exponential function as half-iteration of the exponential, i.e. a function φ such that φ(φ(z)) = exp(z).; although this superexponential is not real and cannot be considered as tetration
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...
.
Kneser was a student of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. He was an advisor of a number of notable mathematicians, including Reinhold Baer
Reinhold Baer
Reinhold Baer was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings....
.
Hellmuth Kneser was a member of the NSDAP and also the SA
Sturmabteilung
The Sturmabteilung functioned as a paramilitary organization of the National Socialist German Workers' Party . It played a key role in Adolf Hitler's rise to power in the 1920s and 1930s...
. In July 1934 he wrote to Ludwig Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.-Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...
a short note supporting his anti-semitic views and stating: "May God grant German science a unitary, powerful and continued political position."