Théophile Lepage

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Théophile Lepage
Born(1901-03-24)24 March 1901
Limburg
Died 1 April 1991(1991-04-01) (aged 90)
Verviers
Nationality Belgian
Alma mater Université libre de Bruxelles
Known for Congruence of Lepage
Calculus of variations
Lepagian forms
Scientific career
Fields Mathematics
Doctoral advisor Théophile de Donder

Théophile Lepage (24 March 1901 – 1 April 1991) was a Belgian mathematician.

Mathematician person with an extensive knowledge of mathematics

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Contents

Biography

Théophile Henri Joseph Lepage, better known as Théophile Lepage, was born in Limburg on March 24, 1901. Together with Alfred Errera he founded the seminar for mathematical analysis at the ULB. This seminar played an important role in the flourishing of the department of mathematics at this university. [1] He was professor of mathematics at the University of Liège from 1928 till 1930. He taught differential and integral calculus at the ULB from 1931 till 1956 and higher analysis from 1956 till 1971.

Limbourg Municipality in French Community, Belgium

Limbourg or Limbourg-sur-Vesdre is a city located in the province of Liège, Wallonia, Belgium.

Alfred Errera was a Belgian mathematician.

Université libre de Bruxelles French-speaking university in Brussels, Belgium

The Université libre de Bruxelles, abbreviated ULB, is a French-speaking private research university in Brussels, Belgium.

For 43 years he was a member of the Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. On June 5, 1948, he was nominated a corresponding member and on June 9, 1956 an effective member of the Académie. In 1963 he became president of the Académie and director of the Klasse Wetenschappen. He was also active in the Belgisch Wiskundig Genootschap.

The Royal Academy of Science, Letters and Fine Arts of Belgium is the independent learned society of science and arts of the French Community of Belgium. It is also called in shorthand Royal Academy of Belgium (ARB) or La Thérésienne from Maria Theresa. The Dutch-speaking counterpart for the Flemish Community in Belgium is called Royal Flemish Academy of Belgium for Science and the Arts. In 2001 both academies founded a joint association for the purpose of promoting science and arts on an international level: The Royal Academies for Science and the Arts of Belgium (RASAB). All three institutions are located in the same building, the Academy Palace in Brussels.

He died in Verviers on April 1, 1991.

His mathematical work

At the ULB, the ideas and the enthusiasm of Théophile de Donder formed the foundation of a flourishing mathematical tradition. Thanks to student Théophile Lepage, external differential calculus acquired one of the most helpful methods introduced in mathematics during the 20th century, and one for which De Donder was a pioneer, presenting new applications in the resolution of a classical problemthe partial differential equation of Monge-Ampère and in the synthesis of the methods of Théophile de Donder, Hermann Weyl and Constantin Carathéodory into a calculus of variations of multipal integrals.

Théophile de Donder Belgian physicist

Théophile Ernest de Donder was a Belgian mathematician and physicist famous for his work in developing correlations between the Newtonian concept of chemical affinity and the Gibbsian concept of free energy.

In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivatives of u. The independent variables (x,y) vary over a given domain D of R2. The term also applies to analogous equations with n independent variables. The most complete results so far have been obtained when the equation is elliptic.

Hermann Weyl German mathematician

Hermann Klaus Hugo Weyl, was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.

Thanks to the use of differential geometry, it is possible to avoid long and boring calculations. The results of Lepage were named in reference works. His methods are still inspiring contemporary mathematicians: Boener and Sniatycki talked about the congruence of Lepage; not so long ago, Demeter Krupka, introducedbeside the eulerian forms which correspond to the classical equations of the calculus of variations of Eulerthe so-called lepagian forms [2] or equivalents of Lepage in equations of variations on fiber spaces.

Differential geometry branch of mathematics

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Fiber bundle continuous surjection satisfying a local triviality condition

In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E and a product space is defined using a continuous surjective map

We also have Lepage to thank for interesting results concerning linear representations of the symplectic group, and more specifically Lepage's dissolution of an outer potency of the product of an even number of duplicates of a complex surface.

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

Symplectic group The group of matrices preserving a non-degenerate alternating quadratic form

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n). The latter is called the compact symplectic group. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the matrices used to represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n.

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References

  1. Mawhin, Jean, in: Robert Halleux, Geert Vanpaemel, Jan Vandersmissen en Andrée Despy-Meyer (eds.), (2001), Geschiedenis van de wetenschappen in België 1815-2000, 1, Brussel: Dexia/La Renaissance du livre, p. 71 and p. 75
  2. D. Krupka (1977). "A map associated to the Lepagian forms on the calculus of variations in fibred manifolds". Czechoslovak Mathematical Journal. 27 (1): 114–117,118.