Areas of interest:
Geometric Function Theory, Spaces of Analytic Functions and Operators Acting on Them, Approximation Theory.

Specific problems and topics I have been studying and/or am studying now:

• Hyperbolic, Schwarzian, and angular derivative of holomorphic self-maps of the disk
  
• Commuting analytic self-maps of the disk and iteration of self-maps of the disk
• (Weighted) composition operators on spaces of analytic functions: norms, isometries, adjoints, spectral theory
  
• Norms and essential norms of composition operators acting on Bergman spaces
• Integrability of derivatives of conformal maps and growth of intergral means
• Integrability of derivatives of Blaschke products; relationship with interpolating sequences
• Uniformly discrete sequences in the disk, Nyquist/Seip densities, interpolation and sampling
• Dirichlet-type spaces (especially analytic Besov spaces, etc.) in the disk
  
• Planar domains related to the images of the disk by conformal maps; univalent interpolation
  
• Superposition operators on spaces of Dirichlet type and other spaces
  
• Applications of inequalities of the Trudinger-Moser type in Complex Analysis
• General theory of Hardy and Bergman spaces
  
• Toeplitz operators on Bergman and Hardy spaces: range and kernel, and zero products 
  
• Analytic Toeplitz operators (pointwise multiplication operators)
  
• The Berezin transform and its applications
  
• Nonlinear problems (maximizing Rayleigh-type quotients) and weighted Bergman spaces
• Extremal problems for linear functionals in spaces of analytic functions
• Extremal problems for non-vanishing functions in spaces of analytic functions (including the Krzyz conjecture) 
• Taylor coefficients of functions in Bergman spaces and coefficient multipliers 
• Hilbert matrix as an operator on function spaces
• Hardy spaces of Dirichlet series