Charles Loewner
Encyclopedia
Charles Loewner was an American
mathematician
. His name was Karel Löwner in Czech and Karl Löwner in German.
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. He worked at the University of Berlin, University of Prague
, Louisville University, Brown University
, Syracuse University
and eventually at Stanford University
. His students include Lipman Bers
, Roger A. Horn, Adriano Garsia
, and P. M. Pu.
, to the effect that every metric on the 2-torus satisfies the optimal inequality
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
.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
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....
. His name was Karel Löwner in Czech and Karl Löwner in German.
Loewner received his Ph.D. from the University of Prague
Charles University in Prague
Charles University in Prague is the oldest and largest university in the Czech Republic. Founded in 1348, it was the first university in Central Europe and is also considered the earliest German university...
in 1917 under supervision of Georg Pick
Georg Alexander Pick
Georg Alexander Pick was an Austrian mathematician. He was born to Josefa Schleisinger and Adolf Josef Pick. He died in the Theresienstadt concentration camp. Today he is best known for Pick's formula for determining the area of lattice polygons...
.
One of his central mathematical contributions is the proof of the Bieberbach conjecture
De Branges' theorem
In complex analysis, the Bieberbach conjecture or de Branges's theorem, posed by and proven by , states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane....
in the first highly nontrivial case of the third coefficient. He worked at the University of Berlin, University of Prague
Charles University in Prague
Charles University in Prague is the oldest and largest university in the Czech Republic. Founded in 1348, it was the first university in Central Europe and is also considered the earliest German university...
, Louisville University, Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
, Syracuse University
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...
and eventually at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
. His students include Lipman Bers
Lipman Bers
Lipman Bers was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.-Biography:...
, Roger A. Horn, Adriano Garsia
Adriano Garsia
Adriano Mario Garsia is an Italian American mathematician, a leading expert in combinatorics, representation theory, and algebraic geometry, a student of Charles Loewner. He has made many deep contributions to representation theory, symmetric functions and algebraic combinatorics, which exerted a...
, and P. M. Pu.
Loewner's torus inequality
In 1949 Loewner proved his torus inequalityLoewner's torus inequality
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.-Statement:...
, to the effect that every metric on the 2-torus satisfies the optimal inequality
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
Hexagonal lattice
The hexagonal lattice or equilateral triangular lattice is one of the five 2D lattice types.Three nearby points form an equilateral triangle. In images four orientations of such a triangle are by far the most common...
spanned by the cube roots of unity in
.Book by Loewner
- Loewner, C.: Theory of continuous groups. Notes by H. Flanders and M. Protter. Mathematicians of Our Time 1, The MIT Press, Cambridge, Mass.-London, 1971.