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