| Charles Loewner | |
| Birth Date: | 29 May 1893 |
| Birth Place: | Lány, Bohemia |
| Death Place: | Stanford, California |
| Nationality: | American |
| Fields: | Mathematics |
| Workplaces: | Stanford University Syracuse University University of Prague |
| Alma Mater: | Karl-Ferdinands-Universität |
| Doctoral Advisor: | Georg Alexander Pick |
| Doctoral Students: | Lipman Bers William J. Firey Adriano Garsia Roger Horn Pao Ming Pu |
| Known For: | Operator monotone function Systolic geometry Loewner equation Loewner order Loewner's torus inequality Loewner–Heinz theorem |
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. 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.[1] [2]
Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick.One of his central mathematical contributions is the proof of the Bieberbach conjecture 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; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, Adriano Garsia, and P. M. Pu.
In 1949 Loewner proved his torus inequality, to the effect that every metric on the 2-torus satisfies the optimal inequality
\operatorname{sys}2\leq
| 2 | |
| \sqrt{3 |
where sys is its systole. 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 spanned by the cube roots of unity in
C
The Loewner matrix (in linear algebra) is a square matrix or, more specifically, a linear operator (of real
C1
n
n x n
Let
f
(a,b)
For any
s,t\in(a,b)
f
s,t
f[1](s,t)=\begin{cases}\displaystyle
| f(s)-f(t) | |
| s-t |
,&ifs ≠ t\\ f'(s),&ifs=t \end{cases}
t1,\ldots,tn\in(a,b)
Lf(t1,\ldots,tn)
f
(t1,\ldots,tn)
n x n
(i,j)
f[1](ti,tj)
In his fundamental 1934 paper, Loewner proved that for each positive integer
n
f
n
(a,b)
Lf(t1,\ldots,tn)
t1,\ldots,tn\in(a,b)
f
n
(a,b)
n
f
"During [Loewner's] 1955 visit to Berkeley he gave a course on continuous groups, 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 and Murray H. Protter "decided to revise and correct the original lecture notes and make them available in permanent form."[6] Charles Loewner: Theory of Continuous Groups (1971) was published by The MIT Press,[7] and re-issued in 2008.[8]
In Loewner's terminology, if
x\inS
S
x
ak{g},
ak{g},
J(\overset{u}{v})
A reviewer said, "The reader is helped by illuminating examples and comments on relations with analysis and geometry."[9]
À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260.