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Tuesday, April 16, 2019

**Abstract:** It is a well-known fact in the theory of several complex variables that a function is holomorphic if and only if it is holomorphic in each variable separately. This result goes back to Hartogs. It is natural to consider a boundary version of Hartogs’ theorem. The general problem is to take a boundary function and ask if holomorphic extensions on some families of complex curves are enough to guarantee an extension which is holomorphic in all variables simultaneously. We will talk about the known results on the subject and show some new results obtained in collaboration with M. Fassina and S. Pinton for the special case of the unit ball in ${\mathbb C}^n$.