Charles Loewner (29 May 1893 – 8 January 1968) was an American
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 ...
. His name was Karel Löwner in Czech and Karl Löwner in German.
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner.
Loewner received his Ph.D. from the
University of Prague in 1917 under supervision of
Georg Pick
Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Theresienstadt concentration camp. Toda ...
.
One of his central mathematical contributions is the proof of the
Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was ...
in the first highly nontrivial case of the third coefficient. The technique he introduced, the
Loewner differential equation, has had far-reaching implications in
geometric function theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.
Topics in geometric function theory
The following are some of the most important topics in ge ...
; it was used in the final solution of the Bieberbach conjecture by
Louis de Branges in 1985. Loewner worked at the
University of Berlin
The Humboldt University of Berlin (german: link=no, Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick Will ...
,
University of Prague,
University of Louisville
The University of Louisville (UofL) is a public research university in Louisville, Kentucky. It is part of the Kentucky state university system. When founded in 1798, it was the first city-owned public university in the United States and one of ...
,
Brown University,
Syracuse University
Syracuse University (informally 'Cuse or SU) is a Private university, private research university in Syracuse, New York. Established in 1870 with roots in the Methodist Episcopal Church, the university has been nonsectarian since 1920. Locate ...
and eventually at
Stanford University. His students include
Lipman Bers
Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also k ...
,
Roger Horn
Roger Alan Horn (born January 19, 1942) is an American mathematician specializing in matrix analysis. He was research professor of mathematics at the University of Utah. He is known for formulating the Bateman–Horn conjecture with Paul T. Batem ...
,
Adriano Garsia
Adriano Mario Garsia (born 20 August 1928) is a Tunisian-born Italian American mathematician who works in analysis, combinatorics, representation theory, and algebraic geometry. He is a student of Charles Loewner and has published work on represe ...
, and
P. M. Pu Pao Ming Pu (the form of his name he used in Western languages, although the Wade-Giles transliteration would be Pu Baoming; ; August 1910 – February 22, 1988), was a mathematician born in Jintang County, Sichuan, China..
He was a student of ...
.
Loewner's torus inequality
In 1949 Loewner proved his
torus inequality, to the effect that every metric on the 2-torus satisfies the optimal inequality
:
where sys is its
systole
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ' ...
. The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called ''equilateral torus'', i.e. torus whose group of deck transformations is precisely the
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
spanned by the cube roots of unity in
.
Loewner matrix theorem
The Loewner matrix (in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
) is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are ofte ...
or, more specifically, a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
(of real
functions) associated with 2 input parameters consisting of (1) a real
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
function on a subinterval of the real numbers and (2) an
-dimensional
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an
matrix.
Let
be a real-valued function that is continuously differentiable on the
open interval .
For any
define the divided difference of
at
as
:
.
Given
, the Loewner matrix
associated with
for
is defined as the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
whose
-entry is
.
In his fundamental 1934 paper, Loewner proved that for each positive integer
,
is
-monotone on
if and only if
is
positive semidefinite for any choice of
.
[ Most significantly, using this equivalence, he proved that is -monotone on for all if and only if is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane. See '']Operator monotone function In linear algebra, the operator monotone function is an important type of real-valued function, first described by Charles Löwner in 1934. It is closely allied to the operator concave and operator concave functions, and is encountered in operator ...
''.
Continuous groups
"During oewner's1955 visit to Berkeley he gave a course on continuous group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
s, and his lectures were reproduced in the form of duplicated notes. Loewner planned to write a detailed book on continuous groups based on these lecture notes, but the project was still in the formative stage at the time of his death." Harley Flanders
Harley M. Flanders (September 13, 1925 – July 26, 2013) was an American mathematician, known for several textbooks and contributions to his fields: algebra and algebraic number theory, linear algebra, electrical networks, scientific computing.
...
and Murray H. Protter
Murray Harold Protter (February 13, 1918 – May 1, 2008) was an American mathematician and educator, known for his contributions to the theory of partial differential equations, as well as his well-selling textbooks in Calculus.
Protter earned a ...
"decided to revise and correct the original lecture notes and make them available in permanent form." ''Charles Loewner: Theory of Continuous Groups'' (1971) was published by The MIT Press
The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States). It was established in 1962.
History
The MIT Press traces its origins back to 1926 when MIT publish ...
, and re-issued in 2008.
In Loewner's terminology, if and a group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphi ...
is performed on , then is called a ''quantity'' (page 10). The distinction is made between an abstract group and a realization of in terms of linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s that yield a group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used t ...
. These linear transformations are Jacobians denoted (page 41). The term ''invariant density'' is used for the Haar measure, which Loewner attributes to Adolph Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...
(page 46). Loewner proves that compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural ge ...
s have equal left and right invariant densities (page 48).
A reviewer said, "The reader is helped by illuminating examples and comments on relations with analysis and geometry."Deane Montgomery
Deane Montgomery (September 2, 1909 – March 15, 1992) was an American mathematician specializing in topology who was one of the contributors to the final resolution of Hilbert's fifth problem in the 1950s. He served as President of the Americ ...
See also
* Löwner-John ellipsoid
* Schramm–Loewner evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional ...
* Loop-erased random walk
In mathematics, loop-erased random walk is a model for a random path (graph theory), simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a ...
* Systoles of surfaces
References
* Berger, Marcel: À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260.
*Loewner, Charles; Nirenberg, Louis: Partial differential equations invariant under conformal or projective transformations. Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York, 1974.
External links
Stanford memorial resolution
*
{{DEFAULTSORT:Loewner, Charles
1893 births
1968 deaths
20th-century American mathematicians
Czech mathematicians
Mathematical analysts
Jewish scientists
Stanford University Department of Mathematics faculty