Salomon Bochner
Encyclopedia
Salomon Bochner was an American mathematician
of Austrian-Hungarian
origin, known for wide-ranging work in mathematical analysis
, probability theory
and differential geometry.
(near Kraków
), then Austria-Hungary, now Poland
. Fearful of a Russian invasion in Galicia at the beginning of World War I
in 1914, his family moved to Germany
, seeking greater security. Bochner was educated at a Berlin
gymnasium
(secondary school), and then at the University of Berlin. There, he was a student of Erhard Schmidt
, writing a dissertation involving what would later be called the Bergman kernel
. 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
from 1924 to 1933. His academic career in Germany ended after the Nazis came to power
in 1933, and he left for a position at Princeton University
. He died in Houston, Texas
. He was an Orthodox Jew.
s, simplifying the approach of Harald Bohr
by use of compactness and approximate identity
arguments. In 1933 he defined the Bochner integral
, as it is now called, for vector-valued functions. Bochner's theorem
on Fourier transform
s appeared in a 1932 book. His techniques came into their own as Pontryagin duality
and then the representation theory of locally compact group
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
behaves under rotations.
In differential geometry, Bochner's formula on curvature
from 1946 was most influential. Joint work with Kentaro Yano (1912–1993) led to the 1953 book Curvature and Betti Numbers. It had broad consequences, for the Kodaira vanishing theory, representation theory
, and spin manifolds. Bochner also worked on several complex variables (the Bochner–Martinelli formula and the book Several Complex Variables from 1948 with W. T. Martin).
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....
of Austrian-Hungarian
Austria-Hungary
Austria-Hungary , more formally known as the Kingdoms and Lands Represented in the Imperial Council and the Lands of the Holy Hungarian Crown of Saint Stephen, was a constitutional monarchic union between the crowns of the Austrian Empire and the Kingdom of Hungary in...
origin, known for wide-ranging work in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and differential geometry.
Life
He was born into a Jewish Family in PodgórzePodgórze
Podgórze is a district of Kraków, Poland, situated on the right bank of the Vistula River. Initially a village at the foot of Lasota Hill was granted city status by the Austrian Emperor Joseph II in 1784 and has become Royal Free City of Podgorze...
(near Kraków
Kraków
Kraków also Krakow, or Cracow , is the second largest and one of the oldest cities in Poland. Situated on the Vistula River in the Lesser Poland region, the city dates back to the 7th century. Kraków has traditionally been one of the leading centres of Polish academic, cultural, and artistic life...
), then Austria-Hungary, now Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
. Fearful of a Russian invasion in Galicia at the beginning of World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
in 1914, his family moved to 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...
, seeking greater security. Bochner was educated at a Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
gymnasium
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...
(secondary school), and then at the University of Berlin. There, he was a student of Erhard Schmidt
Erhard Schmidt
Erhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905...
, writing a dissertation involving what would later be called the Bergman kernel
Bergman kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cn....
. 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
Ludwig Maximilians University of Munich
The Ludwig Maximilian University of Munich , commonly known as the University of Munich or LMU, is a university in Munich, Germany...
from 1924 to 1933. His academic career in Germany ended after the Nazis came to power
Machtergreifung
Machtergreifung is a German word meaning "seizure of power". It is normally used specifically to refer to the Nazi takeover of power in the democratic Weimar Republic on 30 January 1933, the day Hitler was sworn in as Chancellor of Germany, turning it into the Nazi German dictatorship.-Term:The...
in 1933, and he left for a position at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. He died in Houston, Texas
Houston, Texas
Houston is the fourth-largest city in the United States, and the largest city in the state of Texas. According to the 2010 U.S. Census, the city had a population of 2.1 million people within an area of . Houston is the seat of Harris County and the economic center of , which is the ...
. He was an Orthodox Jew.
Mathematical work
In 1925 he started work in the area of almost periodic functionAlmost 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 Harald Bohr and later generalized by Vyacheslav Stepanov,...
s, simplifying the approach of Harald Bohr
Harald Bohr
Harald August Bohr was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr...
by use of compactness and approximate identity
Approximate identity
In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring that acts as a substitute for an identity element....
arguments. In 1933 he defined the Bochner integral
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:...
, as it is now called, for vector-valued functions. Bochner's theorem
Bochner's theorem
In mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line.- Background :...
on Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s appeared in a 1932 book. His techniques came into their own as Pontryagin duality
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
and then the representation theory of locally compact group
Locally compact group
In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
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
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
behaves under rotations.
In differential geometry, Bochner's formula on curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
from 1946 was most influential. Joint work with Kentaro Yano (1912–1993) led to the 1953 book Curvature and Betti Numbers. It had broad consequences, for the Kodaira vanishing theory, representation theory
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 studiesmodules over these abstract algebraic structures...
, and spin manifolds. Bochner also worked on several complex variables (the Bochner–Martinelli formula and the book Several Complex Variables from 1948 with W. T. Martin).