Kharkiv State University
What was the name of the employer Aleksei Pogorelov work for before Jan 1955?
Aleksei Pogorelov Aleksei Vasilevich Pogorelov ( , ; March 2 , 1919 – December 17 , 2002 ) , was a Soviet and Ukrainian mathematician . Specialist in the field of convex and differential geometry , geometric PDEs and elastic shells theory , the author of the novel school textbook on geometry and university textbooks on analytical geometry , on differential geometry , and on foundations of geometry . Pogorelovs uniqueness theorem and the Alexandrov–Pogorelov theorem are named after him . Biography . Born March 3 , 1919 in Korocha , Kursk Governorate ( now Belgorod region ) in a peasant family . In 1931 , because of the collectivization , the parents of A.V . Pogorelov escaped from the village to Kharkiv , where his father become a worker at the construction of the Kharkiv tractor plant . In 1935 , A.V . Pogorelov won the first prize at the Mathematical Olympiad in Kharkiv State University . After high school graduation in 1937 , he entered the mathematical department of the Kharkiv State University . He was the best student at the department . In 1941 , after involving of the Soviet Union into the Second World War , Aleksei Vasilevich was sent for 11 months study to N.Y . Zhukovsky Air Force Engineering Academy . During his studies , the students periodically were sent for several months to the front as technicians for the airplane service . After the Red Army Victory over Nazi near Moscow , the training continued for a full term . After academy graduation , he worked at N.Y . Zhukovsky Central Aero-hydrodynamic Institute ( TsAGI ) as a design engineer . The desire to complete university education and specialize in geometry professionally led A.V . Pogorelov to Moscow State University . By recommendation of I.G . Petrovsky ( Dean of the Mechanics and Mathematics Department ) and a well-known geometer V.F . Kagan , Aleksei Vasilevich met A.D . Aleksandrov – the founder of the theory of non-smooth convex surfaces . There were many new questions concerning this theory . Aleksandr Danilovich proposed to give an answer to one of them to A.V . Pogorelov . In a year the problem was solved and A.V . Pogorelov was enrolled to the graduate school of the Mechanics and Mathematics Department of Moscow State University . Nikolai Efimov became his scientific advisor on topics of Aleksandrov theory . After defending his Ph.D . thesis in 1947 , he was demobilized and moved to Kharkiv , where he started to work at the Institute of Mathematics of Kharkov State University and the Geometry Department of the university . In 1948 he defended his doctoral thesis . In 1951 he became the Corresponding Member of the Academy of Sciences of Ukraine , in 1960 he became the Corresponding member of the USSR Academy of Sciences ( Division of Physical and Mathematical Sciences ) . In 1961 he became an Academician of the Academy of Sciences of Ukraine . In 1976 , he became an Academician of the USSR Academy of Sciences ( Mathematics Division ) . From 1950 to 1960 he was the Head of the Geometry Department at Kharkiv State University . From 1960 to 2000 he was the Head of the Geometry Division at the Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine . Since 2000 he lived in Moscow and worked at the Steklov Mathematical Institute . He died on December 17 , 2002 and was buried in Moscow at the Nikolo-Arkhangelsk cemetery . In 2015 , one of the streets in Kharkiv was named after Academician A.V . Pogorelov . In 2007 , National Academy of Sciences of Ukraine founded the Pogorelov Award for the achievements in the field of geometry and topology . One of the asteroids is named after A.V . Pogorelov : . Awards . - The Stalin Prize of the Second Level ( 1950 ) for works on the theory of convex surfaces , presented in the paper Uniquely Definition of Convex Surfaces and in a series of papers published in Proceedings of the USSR Academy of Sciences ( 1948-1949 ) - Lenin Prize ( 1962 ) – for the results in geometry in the large - Lobachevsky International Prize ( 1959 ) - for the paper Some questions of geometry in the large in Riemannian space - The Krylov Prize of the Academy of Sciences of the Ukrainian SSR ( 1973 ) - The State Prize of the Ukrainian SSR ( 1974 ) - N . N . Bogolubov Prize of the NAS of Ukraine ( 1998 ) - The State Prize of Ukraine ( 2005 ) - Two Lenin Orders - Order of the Labor Banner - Order of the Patriotic War of the II degree ( 06.04.1985 ) Scientific interests . By the beginning of the 20th century , the methods for solving of local problems related to regular surfaces were developed . By the thirties , there were developed the methods for solving the problems in geometry in the large . These methods were related mainly to the theory of partial differential equations . Mathematicians were helpless when surfaces were non-smooth ( for example , with conical points , ribbed points , etc. ) and when the intrinsic geometry was given not by a smooth positive definite quadratic form , but simply by a metric space of a fairly general form . A breakthrough in the study of non-smooth metrics and non-smooth surfaces was made by an outstanding geometer A.D . Aleksandrov . He developed the theory of metric spaces of non-negative curvature , so-called Aleksandrov metric spaces . As a special case , the theory covered the intrinsic geometry of general convex surfaces , that is boundaries of convex bodies . Aleksandrov studied connections between the intrinsic and extrinsic geometries of general convex surfaces . He proved that every metric of non-negative curvature given on a two-dimensional sphere ( including non-smooth metrics , so-called inner metrics ) can be isometrically immersed into the three-dimensional Euclidean space in a form of a closed convex surface , but the answers to the following fundamental questions were unknown : 1 . is this immersion unique up to rigid motion ? 2 . if the metric given on the sphere is a regular one and of positive Gaussian curvature , is it true then that the surface with this metric is regular ? 3 . G . Minkowski proved an existence theorem for a closed convex surface with the Gaussian curvature given as a function of a unit normal under some natural condition on this function ; the open question was : if the function is regular on a sphere , is the surface regular itself ? After solving these problems , the theory created by Aleksandrov would have received “full citizenship” in mathematics and could be applied also in the classical regular case . Each of these 3 questions was answered positively by A.V . Pogorelov . Using synthetic geometric methods , he developed geometric methods to obtain priori estimates for solutions of Monge-Ampère equations . On the one hand , he used these equations to solve geometric problems ; on the other hand , based on geometric reasons , he constructed a generalized solution of a Monge-Ampère equation and then proved its regularity for a regular right-hand side of the equation . In fact , in these pioneering works A.V . Pogorelov laid the foundation of the field of geometric analysis . He proved the following fundamental results : 1 . Let F and F be two closed convex isometric surfaces in the three-dimensional Euclidean space or in a spherical space . Then the surfaces coincide up to the rigid motion . 2 . A closed convex surface in a space of constant curvature is rigid outside flat domains on it . This means that the surface admits only trivial infinitesimal bendings . 3 . If the metric of a convex surface is regular of regularity С , k≥2 , in a space of constant curvature К* and the Gaussian curvature of the surface satisfies К>К* , then the surface is С . For domains on convex surfaces assertions 1 ) and 2 ) are false . The local and global properties of surfaces are significantly different . By proving assertion 1 ) A.V . Pogorelov completed the solution of the problem open for more than a century . The first result in this direction was obtained by Cauchy for closed convex polyhedra in 1813 . The theorems proved by Pogorelov formed the basis for his nonlinear theory of thin shells . This theory is concerned with those elastic states of the shell which differ significantly comparing to the original form . Under such deformations , the middle surface of a thin shell undergoes bending with preservation of the metric . This makes it possible , by using theorems proved by Pogorelov for convex surfaces , to investigate loss of stability and the over critical elastic state of convex shells under a given strain . Such shells are the most common elements of modern designs . Results 1 ) and 2 ) were generalized for regular surfaces in a Riemannian space . In addition , the Weyl problem for Riemannian space was solved : it was proved that a regular metric of Gaussian curvature greater than some constant c on a two-dimensional sphere can be isometrically immersed into a complete three-dimensional Riemannian space of curvature <c in a form of a regular surface . Studying the methods developed in the proof of this result , the Abel Prize laureate M . Gromov introduced the concept of pseudoholomorphic curves , which are the main tool in modern symplectic geometry . A closed convex hypersurface is uniquely defined not only by the metric but also by the Gaussian curvature as a function of unit normals . Moreover , the hypersurface is uniquely determined up to a parallel transport . This was proved by G . Minkowski . But is the hypersurface regular under the condition that the Gaussian curvature K ( n ) is a regular function of a unit normal ? Pogorelov proved that if positive function K ( n ) belongs to the class С , k≥3 , then the support function will be of regularity class С , 0<v<1 . The hardest part of the proof of the theorem was to obtain a priori estimates for the derivatives of the support function of a hypersurface up to third order inclusively . Pogorelovs method of a priori estimates was used by S.-T . Yau to obtain a priori estimates for solutions of complex Monge-Ampere equations . This was the main step in the proof of the existence of Calabi-Yao manifolds , which play an important role in theoretical physics . A Monge-Ampère equation has the form A priori estimates in the Minkowski problem are a priori for the solution of the Monge-Ampère equation with the function At that time there was no approach to studying this completely nonlinear equation . A . V . Pogorelov has created the theory of the Monge-Ampère equation by using the geometric methods . First , going from polyhedra , he proved the existence of generalized solutions under natural conditions on the right-hand side . After that he has found the a priori estimates for the derivatives up to the third order inclusively for the regular solutions . Using the a priori estimates , he has proved the regularity of strictly convex solutions , the existence of solutions of the Dirichlet problem and their regularity . The Monge-Ampère equation is an essential component of the Monge-Kantorovich transport problem ; it is used in conformal , affine , Kähler geometries , in meteorology and in financial mathematics . A.V . Pogorelov once said about the Monge-Ampère equation : this is a great equation with which I had the honor to work . One of the most conceptual works of A . V . Pogorelov refers to the cycle of works about smooth surfaces of bounded external curvature . A.D . Aleksandrov created a theory of general metric manifolds that naturally generalize Riemannian manifolds . In particular , he introduced the class of two-dimensional manifolds of bounded curvature . They exhaust the class of all metrized two-dimensional manifolds that admit , in a neighborhood of each point , a uniform approximation by Riemannian metrics with absolute integral curvature ( i.e. , the integral of the module of Gaussian curvature ) bounded in aggregate . Naturally , the question arose about the class of surfaces in three-dimensional Euclidean space carrying such a metric with preservation of connections between the metric and the extrinsic geometry of the surface . Partially answering this question , A.V . Pogorelov introduced the class of С-smooth surfaces with the requirement on the area of a spherical image to be bounded , taking into account the multiplicity of the covering in some neighborhood of each point of the surface . Such surfaces are called surfaces of bounded extrinsic curvature . For such surfaces there is also a very close connection between the intrinsic geometry of the surface and its extrinsic shape : a complete surface with a bounded extrinsic curvature and a nonnegative intrinsic curvature ( not equal to zero ) is either a closed convex surface or an unbounded convex surface ; a complete surface with zero intrinsic curvature and bounded extrinsic curvature is a cylinder . The first work of A . V . Pogorelov on surfaces of bounded extrinsic curvature was published in 1953 . In 1954 , J . Nash published the paper on С-isometric immersions , which was improved by N . Kuiper in 1955 . It follows from these studies that a Riemannian metric defined on a two-dimensional manifold , under very general assumptions , admits a realization on a С-smooth surface in a three-dimensional Euclidean space . Moreover , this realization is carried out as freely as a topological immersion into the space of the manifold on which the metric is given . Hence it is clear that for С-surfaces , even with a good intrinsic metric , it is impossible to preserve the connections between the intrinsic and extrinsic curvatures . Even in case if a С-surface carries a regular metric of positive Gaussian curvature , then this does not imply the local convexity of the surface . This emphasizes the naturalness of the class of surfaces of bounded external curvature introduced by A . V . Pogorelov . A . V . Pogorelov solved Hilberts fourth problem , set by D . Hilbert at the II International Congress of Mathematicians in Paris in 1900 . He found all , up to isomorphism , realizations of the systems of axioms of classical geometries ( Euclid , Lobachevsky and elliptic ) if one omits the congruence axioms containing the concept of angle and supplement these systems with the axiom of triangle inequality . A . V . Pogorelov was one of the first who has proposed ( in 1970 ) a new idea in the construction of a cryoturbogenerator with superconducting field winding and took an active part in technical calculations and creation of corresponding industrial samples . Selected publications . - Geometry [ translated from the Russian by Leonid Levant , Aleksandr Repyev and Oleg Efimov. ] . Moscow : Mir Publishers ( 1987 ) . . .