Research areas that I am interested in:

- Geometric Function Theory
- Real and Harmonic Analysis
- Optimal Mass Transportation
- Nonlinear elliptic PDE
- Geometric Measure Theory
- Complex Analysis
- Transport equation
- Non-smooth vector fields
- Euler equation
- Inverse Boundary Value Problems
- Geometric Analysis
- Gradient flows in Banach spaces

Some keywords:

- quasiconformal mappings and quasiregular mappings;
- Calderón-Zygmund theory;
- apriori estimates for elliptic equations; existence and uniqueness;
- degenerate elliptic operators; commutators;
- function spaces; interpolation; characterizations;
- optimal mass transportation; continuity equation;
- Hamilton-Jacobi equations; viscosity solutions;
- Wardrop equilibrium; transport density; Monge problem;
- Muckenhoupt weights and reverse Hölder inequalities;
- nonlinear elliptic and parabolic PDEs; flows of solutions;
- Fréchet manifolds; curves in the Wasserstein space; surfaces in the Sobolev space;
- calculus of variations; smoothness of minimizers;
- linear transport equation; well posedness; admissible vector fields
- extremal transport problems and nonlinear transport equations
- non-smooth flows; DiPerna-Lions flows
- Bergman projections; truncated spaces; stability;

Some research projects:

- 2 – dimensional manifolds of quasiconformal mappings
- Muckenhoupt weights, quasiconformal mappings, and elliptic operators
- Besov-Lipschitz and Triebel-Lizorkin estimates for nonlinear elliptic equations
- traffic problems in optimal mass transportation; Wardrop equilibrium
- Bergman kernels, Beltrami equations and Schwarz domains
- regularity of viscosity solutions of Hamilton-Jacobi equations
- apriori L^2 bounds for degenerate elliptic equations in divergence form
- well-posedness for the transport equation in non-reflexive Banach spaces
- flows of non-smooth vector fields with exponentially integrable divergence
- vorticity estimates for the Euler equation