Adolf Hurwitz
Encyclopedia
Adolf Hurwitz was a German
mathematician
.
family in Hildesheim
, former Kingdom of Hannover, now Lower Saxony
, Germany
, and died in Zürich
, in Switzerland
. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed. His father, Salomon Hurwitz, was in the manufacturing business but was not particularly well off. Hurwitz's mother, Elise Wertheimer, died when he was only three years old. Hurwitz entered the Realgymnasium Andreanum in Hildesheim in 1868. He was taught mathematics there by Hermann Schubert
. Schubert persuaded Hurwitz's father to allow him to go to university, and arranged for Hurwitz to study with Felix Klein
at Munich. Salomon Hurwitz could not afford to send his son to University, but his friend, Mr Edwards, agreed to help out financially.
, Weierstrass
and Kronecker
, after which he returned to Munich.
In October 1880, Felix Klein moved to the University of Leipzig
. Hurwitz followed him there, and became a doctoral student under Klein's direction, finishing a dissertation on elliptic modular functions in 1881. Following two years at the University of Göttingen, in 1884 he was invited to become an Extraordinary Professor at the Albertus Universität in Königsberg
; there he encountered the young David Hilbert
and Hermann Minkowski
, on whom he had a major influence. Following the departure of Frobenius
, Hurwitz took a chair at the Eidgenössische Polytechnikum Zürich (today the ETH Zürich
) in 1892 (having to turn down a position at Göttingen shortly after ), and remained there for the rest of his life.
Throughout his time in Zürich, Hurwitz suffered from continual ill health, which had been originally caused when he contracted typhoid whilst a student in Munich. He suffered from severe migraines, and then in 1905, his kidneys became diseased and he had one removed.
theory, and used it to prove many of the foundational results on algebraic curve
s; for instance Hurwitz's automorphisms theorem. This work anticipates a number of later theories, such as the general theory of algebraic correspondences, Hecke operator
s, and Lefschetz fixed-point theorem
. He also had deep interests in number theory
. He studied the maximal order theory (as it now would be) for the quaternion
s, defining the Hurwitz quaternion
s that are now named for him.
, Hurwitz met and married Ida Samuel, the daughter of a professor in the faculty of medicine. They had three children.
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....
.
Early life
He was born to a JewishJudaism
Judaism ) is the "religion, philosophy, and way of life" of the Jewish people...
family in Hildesheim
Hildesheim
Hildesheim is a city in Lower Saxony, Germany. It is located in the district of Hildesheim, about 30 km southeast of Hanover on the banks of the Innerste river, which is a small tributary of the Leine river...
, former Kingdom of Hannover, now Lower Saxony
Lower Saxony
Lower Saxony is a German state situated in north-western Germany and is second in area and fourth in population among the sixteen states of Germany...
, 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...
, and died in Zürich
Zürich
Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...
, in Switzerland
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed. His father, Salomon Hurwitz, was in the manufacturing business but was not particularly well off. Hurwitz's mother, Elise Wertheimer, died when he was only three years old. Hurwitz entered the Realgymnasium Andreanum in Hildesheim in 1868. He was taught mathematics there by Hermann Schubert
Hermann Schubert
Hermann Cäsar Hannibal Schubert was a German mathematician.Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen...
. Schubert persuaded Hurwitz's father to allow him to go to university, and arranged for Hurwitz to study with Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
at Munich. Salomon Hurwitz could not afford to send his son to University, but his friend, Mr Edwards, agreed to help out financially.
Educational career
Hurwitz entered the University of Munich in 1877, aged 17. He spent one year there attending lectures by Klein, before spending the academic year 1877-1878 at the University of Berlin where he attended classes by KummerErnst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
, Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
and Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
, after which he returned to Munich.
In October 1880, Felix Klein moved to the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...
. Hurwitz followed him there, and became a doctoral student under Klein's direction, finishing a dissertation on elliptic modular functions in 1881. Following two years at the University of Göttingen, in 1884 he was invited to become an Extraordinary Professor at the Albertus Universität 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...
; there he encountered the young 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...
and Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
, on whom he had a major influence. Following the departure of Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, Hurwitz took a chair at the Eidgenössische Polytechnikum Zürich (today the ETH Zürich
ETH Zurich
The Swiss Federal Institute of Technology Zurich or ETH Zürich is an engineering, science, technology, mathematics and management university in the City of Zurich, Switzerland....
) in 1892 (having to turn down a position at Göttingen shortly after ), and remained there for the rest of his life.
Throughout his time in Zürich, Hurwitz suffered from continual ill health, which had been originally caused when he contracted typhoid whilst a student in Munich. He suffered from severe migraines, and then in 1905, his kidneys became diseased and he had one removed.
Contributions to mathematics
He was one of the early masters of the Riemann surfaceRiemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
theory, and used it to prove many of the foundational results on algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s; for instance Hurwitz's automorphisms theorem. This work anticipates a number of later theories, such as the general theory of algebraic correspondences, Hecke operator
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
s, and Lefschetz fixed-point theorem
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
. He also had deep interests in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. He studied the maximal order theory (as it now would be) for the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s, defining the Hurwitz quaternion
Hurwitz quaternion
In mathematics, a Hurwitz quaternion is a quaternion whose components are either all integers or all half-integers...
s that are now named for him.
Family
In 1884, whilst at KönigsbergKönigsberg
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...
, Hurwitz met and married Ida Samuel, the daughter of a professor in the faculty of medicine. They had three children.
See also
- Hurwitz's theorem (complex analysis)Hurwitz's theorem (complex analysis)In complex analysis, a field within mathematics, Hurwitz's theorem, named after Adolf Hurwitz, roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, then after a while those functions and the limit...
- Riemann–Hurwitz formula
- Hurwitz's theorem (normed division algebras)
- Hurwitz's automorphisms theorem
- Hurwitz's theorem (number theory)Hurwitz's theorem (number theory)In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many rationals m/n such that...
- Hurwitz matrixHurwitz matrix-Hurwitz matrix and the Hurwitz stability criterion:In mathematics, Hurwitz matrix is a structured real square matrix constructed with coefficientsof a real polynomial...
- Hurwitz polynomialHurwitz polynomialIn mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative...
- Hurwitz quaternion orderHurwitz quaternion orderThe Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967...
- Hurwitz zeta function
- First Hurwitz tripletFirst Hurwitz tripletIn the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 . The explanation for this phenomenon is arithmetic...
- Hurwitz determinantHurwitz determinantIn mathematics, Hurwitz determinants were introduced by , who used them to give a criterion for all roots of a polynomial to have negative real part.-Definition:Let us consider a characteristic polynomial P in the variable λ of the form:...