After high school graduation in 1937, he entered the mathematical department of the Kharkiv State University.
In 1941, after the involvement of the Soviet Union in the World War II, Pogorelov was sent for 11 months study to N.Y. Zhukovsky Air Force Engineering Academy.
After academy graduation, he worked at N.Y. Zhukovsky Central Aero-hydrodynamic Institute (TsAGI) as a design engineer.
In a year the problem was solved and Pogorelov was enrolled to the graduate school of the Mechanics and Mathematics Department of Moscow 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.
By the beginning of the 20th century, the methods for solving of local problems related to regular surfaces were developed.
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: 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.
This theory is concerned with those elastic states of the shell which differ significantly comparing to the original form.
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 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. 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. This was the main step in the proof of the existence of Calabi-Yau manifolds, which play an important role in theoretical physics. 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. It follows from these studies that a Riemannian metric defined on a two-dimensional manifold, under very general assumptions, admits a realization on a С1-smooth surface in a three-dimensional Euclidean space. Even in case if a С1-surface carries a regular metric of positive Gaussian curvature, then this does not imply the local convexity of the surface. 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. In 2007, National Academy of Sciences of Ukraine founded the Pogorelov Award for the achievements in the field of geometry and topology.