Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian
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 ...
, known for work in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
,
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
.
Life
He was born into a Jewish family in
Podgórze
Podgórze is a district of Kraków, Poland, situated on the right (southern) bank of the Vistula River, at the foot of Lasota Hill. The district was subdivided in 1990 into six new districts, see present-day districts of Kraków for more details. ...
(near
Kraków
Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 ...
), then Austria-Hungary, now
Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous ...
. Fearful of a Russian invasion in
Galicia at the beginning of
World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
in 1914, his family moved to Germany, seeking greater security. Bochner was educated at a
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
gymnasium (secondary school), and then at the
University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
. There, he was a student of
Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...
,
writing a dissertation involving what would later be called the
Bergman kernel In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space ( RKHS) of all square integrable holomorphic functions on a domain ''D'' in C''n''.
In de ...
. Shortly after this, he left the academy to help his family during the
escalating inflation. After returning to mathematical research, he lectured at the
University of Munich
The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universität München) is a public research university in Munich, Germany. It is Germany's List of universities in Germany, sixth-oldest u ...
from 1924 to 1933. His academic career in Germany ended after the
Nazis came to power
Adolf Hitler's rise to power began in the newly established Weimar Republic in September 1919 when Hitler joined the '' Deutsche Arbeiterpartei'' (DAP; German Workers' Party). He rose to a place of prominence in the early years of the party. Be ...
in 1933, and he left for a position at
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
. He was a visiting scholar at the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
in 1945-48. He was appointed as Henry Burchard Fine Professor in 1959, retiring in 1968. Although he was seventy years old when he retired from Princeton, Bochner was appointed as Edgar Odell Lovett Professor of Mathematics at
Rice University
William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranke ...
and went on to hold this chair until his death in 1982. He became Head of Department at Rice in 1969 and held this position until 1976. He died in
Houston, Texas
Houston (; ) is the most populous city in Texas, the most populous city in the Southern United States, the fourth-most populous city in the United States, and the sixth-most populous city in North America, with a population of 2,304,580 in ...
. He was an
Orthodox Jew
Orthodox Judaism is the collective term for the traditionalist and theologically conservative branches of contemporary Judaism. Theologically, it is chiefly defined by regarding the Torah, both Written and Oral, as revealed by God to Moses on ...
.
Mathematical work
In 1925 he started work in the area of
almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Haral ...
s, simplifying the approach of
Harald Bohr
Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the No ...
by use of
compactness and
approximate identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
Definition
A right approximate ...
arguments. In 1933 he defined the
Bochner integral In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions.
Definition
Let (X, \Sigma, \mu) be a meas ...
, as it is now called, for vector-valued functions.
Bochner's theorem
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a c ...
on
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s appeared in a 1932 book. His techniques came into their own as
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
and then the representation theory of
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
s developed in the following years.
Subsequently, he worked on
multiple Fourier series, posing the question of the
Bochner–Riesz means. This led to results on how the Fourier transform on
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 ...
behaves under rotations.
In differential geometry,
Bochner's formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
Formal statement
If u \colon M \righ ...
on
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
from 1946 was published. Joint work with
Kentaro Yano (1912–1993) led to the 1953 book ''Curvature and Betti Numbers''. It had consequences, for the
Kodaira vanishing theory
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The impl ...
,
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, and
spin manifolds. Bochner also worked on
several complex variable
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
s (the
Bochner–Martinelli formula and the book ''Several Complex Variables'' from 1948 with
W. T. Martin
William Ted Martin (June 4, 1911 – May 30, 2004) was an American mathematician, who worked on mathematical analysis, several complex variables, and probability theory. He is known for the Cameron–Martin theorem and for his 1948 book ''Several ...
).
Selected publications
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2016 reprint*
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2013 reprint*
[2014 reprint]
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See also
* Bochner almost periodic functions
*Bochner–Kodaira–Nakano identity
In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold in terms of its compl ...
* Bochner Laplacian
*Bochner measurable function In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,
...
References
External links
*
*
National Academy of Sciences Biographical Memoir
*
{{DEFAULTSORT:Bochner, Salomon
1899 births
1982 deaths
20th-century German mathematicians
20th-century American mathematicians
Jewish scientists
Differential geometers
Complex analysts
Mathematical analysts
Measure theorists
PDE theorists
Princeton University faculty
Rice University faculty
Polish Orthodox Jews
American Orthodox Jews
Jews from Galicia (Eastern Europe)
Scientists from Berlin
Institute for Advanced Study visiting scholars
Jewish emigrants from Nazi Germany to the United States