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This paper is based on a talk given at the bicentenary celebration of the birth of Niels Henrik Abel held in Oslo in June, 2002. The objectives of the talk were first to recall Abel’s theorem in more or less its original form, secondly to discuss two of the perhaps less well known converses to the theorem, and thirdly to present two (from among the many) interesting issues in modern algebraic geometry...
The problem of constructing algebraic cycles with given properties — e.g., in a given homology class - is one of the longest standing and deepest questions in algebraic geometry. Over the years there have been several variational approaches centered around the question