The report consists of two parts. In the first I will tell about my institute, in which I am the deputy director and head of the department of the theory of differential equations and about its main results.

Fields of scientific research PIAPMM are:

• Methods of nonlinear functional analysis, linear algebra, differential geometry and topology

• Nonclassical problems of theory of differential and integral equations and relativistic mathematical physics

• Mathematical and thermodynamic modelling of deformable bodies in dependence on the structure and interaction between different physical processes

• Mathematical methods for calculation, optimization and prediction of deformability, strength, durability and behavior of mechanical systems

In the second part I will tell about my personal results in the field of mathematical physics, more precisely, in the field of geometrical analysis. Geometrical analysis uses geometrical approaches to solving differential equations and methods of the theory of differential equations for solving geometrical problems. Both areas include General Relativity, which is geometrical theory in terms of partial differential equations.

A typical example is the question about zeros (nodes) of Sen-Witten equation (SWE) which was a subject of investigations from the time of introduction SWE by Edward Witten in proving of Positive Energy Theorem (1981) in connection with question about correlation between tensor and spinor methods in PET proving, and in connection with grounding of the tensor method of prove and also in connection with application of special orthonormal frame (SOF) in numerical relativity. For establishing conditions of nodal sets absence we for the first time pointed (JMP, 2000), that the nodal points of spinorial field λ can be removed by choice of appropriate boundary condition and coefficients in squared SWE, and obtained such conditions. With application of these conditions we obtained also a conclusion about exact form of correlation between SWE and SOF (CQG, 2001). As development and refinement of these results we present our new result.

Theorem 1. Let：

a）everywhere in the bounded domain Ω on the space-like hypersurface ∑t in Petrov type N space-time the dominant energy conditions be fulfilled;

b）the functions Reλ0 or Imλ0, which correspond to non-negative eigenvalue C0，on the boundary ðΩ nowhere equal zero. Then the non-trivial solution λc to SWE does not have the nodal points in the domain Ω on ∑t. 

Corollary. Let on the space-like hypersurface ∑t in Petrov type N space-time Einstein constrains and the dominant energy condition be fulfilled. Then everywhere on ∑t there exists a two-to-one correspondence between Sen-Witten spinor and Sen-Witten orthonormal frame.

This result proves geometrical nature of Witten spinor field and grounded tensorial proof of PET.

On the basis of the experience obtained in the analysis of spinor equations we at first time obtain in an analytical form a class of general and separated solutions of the equations of spinor fields of arbitrary spin, including Maxwell field (which in partial case give Chandrasekhar solution). As corollary, we confirm precisely in the wave optics approach one well-known physical effect in the Kerr field, which describes rotated black holes, and anticipate three new effects. This effects probably can be discovered by China’s Flagship X-Ray Space Observatory, eXTP, launch planned for 2025.