On unity of mathematics

Formally, mathematics is divided into sections where the own problems are posed. Within these sections, for solving problems, research methods are created. This leads to isolation of sections and blocks the creation of new directions. For example, the classical solvability of the Monge–Ampère equation was proved by the elliptic partial differential equation theory methods by 1952, but only in the two-dimensional case (L. Nirenberg et al.). In a multidimensional case, the problem remained open until 1970, when A.V. Pogorelov in a series of works 1970–1973 found a new analytical approach, which in combination with purely geometric methods, led to the proof of the existence and uniqueness of regular solutions to the Dirichlet problem for the multidimensional Monge–Ampère equation. These works motivated the search for purely analytical methods to prove the smoothness of solutions to a wider class of nonlinear equations in the multidimensional case. In 1982, these amazing methods were created in L.C.Evans', N.V. Krylov' and M.V. Safonov's works. Other examples of this type will be given.