Mathematics Colloquia and Seminars

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A one-parameter system of polynomial equations in several variables satisfies "weak approximation" if every solution which is a power series in the parameter can be approximated to arbitrary order by a solution which is a polynomial in the parameter (this is close to "weak approximation" in number theory). According to a conjecture of Hassett and Tschinkel, a system should satisfy weak approximation if and only if for a general choice of the parameter the corresponding system is "rationally connected", i.e., any two solutions are interpolated by a one-parameter family of solutions which are the outputs of a rational function. I will explain the motivation for this conjecture, the evidence due to Hasset-Tschinkel, et al., and new results due to Mike Roth and myself, and to Zhiyu Tian. The new technique is a "local-global" principle for understanding the Hilbert scheme (a parameter space for polynomial solutions of the system).