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I will give a brief overview on Interpolation Problems, a broad area of mathematics whose origin goes back to the beginning of Algebraic Geometry.
I will present the general setting and our contribution to this field via a geometrical approach. We study linear systems of hypersurfaces of a fixed degree passing through a collection of points in general position with assigned multiplicities. We are interested in the dimension of such linear systems. I will explain the Cremona action on r-cycles of the blown up projective space via birational geometry and give applications of this construction to interpolation problems. I will present connections of this dimensionality problem to Gromov-Witten theory. Results in this talk are obtained in collaboration with M.C. Brambilla, R. Miranda and E. Postinghel.