Lev Semenovich Pontryagin.

Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician. He was born in Moscow and lost his eyesight in a primus stove explosion when he was 14. Despite his blindness he was able to become a mathematician due to the help of his mother Tatyana Andreevna who read mathematical books to him. He made major discoveries in a number of fields of mathematics, including the more geometric parts of topology. Image File history File links Pontriagin. ... Image File history File links Pontriagin. ... September 3 is the 246th day of the year (247th in leap years). ... 1908 (MCMVIII) was a leap year starting on Wednesday (link will take you to calendar). ... May 3 is the 123rd day of the year in the Gregorian calendar (124th in leap years). ... 1988 (MCMLXXXVIII) was a leap year starting on Friday of the Gregorian calendar. ... State motto (Russian): Пролетарии всех стран, соединяйтесь! (Transliterated: Proletarii vsekh stran, soedinyaytes!) (Translated: Workers of the world, unite!) Capital Moscow Official language None; Russian (de facto) Government Federation of Soviet republics Area  - Total  - % water 1st before collapse 22,402,200 km² Approx. ... Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... For other uses, see Moscow (disambiguation). ... A small portable stove and its container MSR WindPro with skillet, heat reflector, wind shield and isobutane/propane canister A portable stove is a stove specially designed to be portable and lightweight, as for camping. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

He worked on duality theory for homology while still a student. He went on to lay foundations for the abstract theory of the Fourier transform, now called Pontryagin duality. In topology he posed the basic problem of cobordism theory. This led to the introduction around 1940 of a theory of characteristic classes, now called Pontryagin classes, designed to vanish on a manifold that is a boundary. Moreover, in operator theory a specific subclass of Krein spaces is called Pontryagin spaces. The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... The Fourier transform, named after Joseph Fourier, is a reversible integral transform of one function into another. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. ... In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is twisted — particularly, whether it possesses sections or not. ... In mathematics, the Pontryagin classes are certain characteristic classes. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... The word Boundary has a variety of meanings. ... In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. ... Krein spaces (axiomatics) 1 We consider a linear space with a Q-metric . ...

Later in his career he worked in optimal control theory. His minimum principle is fundamental to modern theory on the subject. He also introduced there the idea of a bang-bang principle, to describe situations where either the maximum 'steer' should be applied to a system, or none. Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimization algorithms. ... Pontryagins minimum principle is used in optimal control theory to find the best possible control for taking a dynamic system from one state to another. ... In optimal control problems, it is sometimes the case that a control is restricted to be between a lower and an upper bound. ...

External links

• Lev Semenovich Pontryagin at the Mathematics Genealogy Project
• O'Connor, John J., and Edmund F. Robertson. "Lev Semenovich Pontryagin". MacTutor History of Mathematics archive.
• Autobiography of Pontryagin (in Russian)