We solve difficult problems of geometric analysis on non-holonomic structures (in other words, on sub-Riemannian manifolds and their generalizations, those are Carnot – Carathéodory spaces). Such structures arise naturally in problems of physics, mechanics, neurobiology, aerodynamics, contact and hyperbolic geometry, harmonic analysis (CR complex boundaries), geometric control theory, the theory of subelliptic equations, etc.
Our main areas of research:
- the structure of (sub)-Riemannian manifolds;
- new connections between topology and curvature of (sub)-Riemannian manifolds;
- application of the theory (sub)-Riemannian manifolds to solving analysis problems, the theory of subelliptic equations, geometric control theory, hydrodynamics and other fields.
Our Laboratory achieved the following results:
- we investigated connections and curvatures of three-dimensional sub-Riemannian spaces;
- we found geodesics and calculated sectional curvatures for complete connected semisimple Lie groups of isometries of symmetric Riemannian manifolds equipped with a natural left-invariant sub-Riemannian metric;
- we calculated the sub-Riemannian curvature of a C2-smooth Euclidean regular curve in the special linear group SL(2,R);
- we obtained upper bounds (exact values in even-dimensional spaces) of the minimum possible multiplicity of a circle which admits a certain non-degenerate continuous deformation;
- we derived the area formula on two-step sub-Lorentzian structures;
- for systems of C1-smooth vector fields, we found a necessary condition for a coordinate system to be privileged;
- we laid the foundations for the quasiconformal analysis of a new two-index scale of spatial maps;
- we obtained conditions for the boundedness of composition operators and multiplication operators on Lebesgue spaces with mixed norm;
- we proved uniform boundedness of solutions to the stationary system of Navier–Stokes equations with a finite Dirichlet integral in the outer plane domain;
- we established a criterion of potentiality for horizontal vector fields;
- we constructed coercive estimates for special differential operators on the Heisenberg group which generalize Korn’s inequality.
Scientific supervisor of the laboratory: Professor S. K. Vodopyanov, head of the Chair of Mathematical Analysis at NSU, head of the Geometric control theory Laboratory at Sobolev Institute of Mathematics.
Scientific project: The grant of the Ministry of Education and Science of the Russian Federation (grant No 1.3087.2017/4.6) entitled «Geometric analysis on (sub)-Riemannian manifolds and applications». The scientific adviser of the project is Professor Vodopyanov S. K.