Title: “Interpolation for polynomials in two variables”
Abstract: I will discuss the following problem: given n general points {p_i} in the plane, and positive integer {m_i}, what is the dimension of the space of polynomials f(x,y) of degree d having multiplicity at least m_i at p_i for each i?  Various conjectures are available for different aspects: Segre’s Conjecture, Nagata’s Conjecture, Segre-Gimigliano-Harbourne-Hirschowitz Conjecture, etc.  One can organize the information with the parameters d, {m_i}, and consider the various cones of interest in the (n+1)-dimensional space that encode those parameters for which the corresponding linear system is nef, is effective, etc.  In recent work with J. Roe and C. Ciliberto we’ve identified interesting rays on the boundary of the Mori cone which are irrational, the first such examples known.

Rick Miranda
Professor of Mathematics and Interim President
Colorado State University