Analytic Number Theory
Extremal problems for polynomials with integer coefficients.
Integer Chebyshev polynomials and constants. Distribution
of conjugate algebraic numbers. Mahler's measure and other polynomial heights. Norms of products and factors of polynomials. Distribution of primes.
Convergence, asymptotics and zero distribution of polynomials and rational functions. Approximation of conformal mappings via the Bergman and the Szego kernel methods.
Extremal length, capacity, harmonic measure and boundary behavior of analytic and harmonic functions. Polynomials and polynomial inequalities.
Orthonormalization methods for numerical approximation of conformal
mappings. Discrete approximations of extremal problems in complex analysis and potential theory. Convergence in numerical methods for systems of linear equations.
Potentials, capacities, equilibrium measures, Riesz decompositions, and applications.
Random polynomials and analytic functions, random matrices. Probabilistic potential theory and number theory.