Charles Loewner
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. 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. 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. 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 formula_2. Loewner matrix theorem. The Loewner matrix (in linear algebra) is a square matrix or, more specifically, a linear operator (of real formula_3 functions) associated with 2 input parameters consisting of (1) a real continuously differentiable function on a subinterval of the real numbers and (2) an formula_4-dimensional vector with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an formula_5  matrix. Let formula_6 be a real-valued function that is continuously differentiable on the open interval formula_7. For any formula_8 define the divided difference of formula_6 at formula_10 as Given formula_12, the Loewner matrix formula_13 associated with formula_6 for formula_15 is defined as the formula_5 matrix whose formula_17-entry is formula_18. In his fundamental 1934 paper, Loewner proved that for each positive integer formula_4, formula_6 is formula_4-monotone on formula_7 if and only if formula_13 is positive semidefinite for any choice of formula_24. Most significantly, using this equivalence, he proved that formula_6 is formula_4-monotone on formula_7 for all formula_4  if and only if formula_6 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". Continuous groups. "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." "Charles Loewner: Theory of Continuous Groups" (1971) was published by The MIT Press, and re-issued in 2008. In Loewner's terminology, if "x" ∈ "S" and a group action is performed on "S", then "x" is called a "quantity" (page 10). The distinction is made between an abstract group formula_30 and a realization of formula_30 in terms of linear transformations that yield a group representation. These linear transformations are Jacobians denoted formula_32 (page 41). The term "invariant density" is used for the Haar measure, which Loewner attributes to Adolph  Hurwitz (page 46). Loewner proves that compact groups 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."