Algebra

• Lie algebras, quantum groups, Poisson geometry, representation theory

Geometry

• Affine differential geometry, geometry of submanifolds in affine space
• Geometry of submanifolds in fiber bundles, geometric properties of vector fields and sections of vector bundles
• Differential geometry of Lie groups and homogeneous spaces, geometry of submanifolds in homogeneous spaces
• Grassmann manifolds and Grassmann image of submanifolds, pseudospherical submanifolds, Bianchi-Bäcklund transformations, minimal surfaces, polyhedra and their flexions, integrable systems
• Foliations of Riemannian manifolds, geometric topology, metric geometry

Partial differential equations and mathematical physics

• Asymptotic behavior of solutions of partial differential equations, dynamical systems
• Differential equations with delay; state-dependent delay, dynamical systems
• Inverse problems of spectral theory for differential equations, integrable nonlinear equations

Theory of functions

• Harmonic analysis, theory of quasicrystals

Functional Analysis

• Theory of Banach spaces
• Operator theory, interpolation problems of analysis

Algebraic and arithmetic questions of the theory of differential and difference equations