Siciak's School of Complex Analysis
   The 'birthday cake' presents the most important directions of complex analysis research in Kraków. Only selected topics are presented, and the presentation is in no way to be considered complete. The material used for preparing 'the cake' was limited to about 500 papers written by the mathematicians from our Institute, published since the late fifties until now. The percentages describe how many of the 500 papers concern a particular topic. For some papers it is not easy to decide which sections they belong to, so the numbers are just approximate. The pieces of the 'pie' are ordered chronologically with respect to the date of publication of the first paper in each section (from 'Extremal Functions', clockwise).
   The bar chart describes which mathematicians from our Institute have contributed to which sections.

  • 1957 Extremal points and functions. Siciak's extremal function.
  • 1961 General theory. Infinite holomorphy. Exceptional sets for functions from Bergman spaces. Separately analytic, harmonic and subharmonic functions. Extension and decomposition theorems. Hörmander operator.
  • 1965 Approximation and interpolation theory. Quasianalytic functions of several complex variables. L-regularity. Approximation and interpolation of entire functions. Continuity criteria for Siciak's extremal function. Pluriregularity of subanalytic sets. Markov's and Bernstein's inequalities. Extension of Whitney jets. Jackson's theorem. Minimal projections.
  • 1968 Analytic continuation. Analytic continuation of harmonic functions. Domains of existence of special families of holomorphic functions, e.g. bounded holomorphic functions, holomorphic functions with polynomial growth. Highly noncontinuable functions.
  • 1979 Polynomial automorphisms. Keller's jacobian conjecture. Reduction procedures. Real jacobian conjecture. Identity sets for polynomial automorphisms.
  • 1980 Algebraic and analytic geometry. Analytic sets with proper projections. Separately algebraic sets. Nash functions. Proper polynomial mappings. Gröbner bases and their applications. Effective methods in algebraic geometry. Calabi-Yau manifolds.
  • 1981 Intersection theory. Continuity of total number of intersection. Bezout and Rouché type theorems. Improper intersections. Separation exponent.
  • 1983 Pluripotential theory. Monge-Ampere operator. Dirichlet problem.
  • 1988 Invariant distances. Carathéodory pseudodistance. Kobayashi pseudodistance. Bergman kernel. Product property. Pluricomplex Green function. Complex geodesics. Hyperbolicity. Completeness.
Marek Jarnicki
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