CAREER: Higher rational connectedness, higher Fano manifolds, and applications

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Rational connectedness, rational simple connectedness, etc., are algebro-geometric analogues of path connectedness, simple connectedness, etc. Just as higher connectedness and higher homotopy groups play an important role in topological obstruction theory, so the algebro-geometric analogues should play an important role in algebro-geometric obstruction theory. In particular, new ideas coming from A1-homotopy theory should resolve the weak approximation problem of Hassett and Tschinkel. There are 3 objectives of the research component. One objective is to develop an algebro-geometric theory of higher rational connectedness analogous to the theory of topological obstruction theory, and with applications to existence of rational sections ofalgebraic fibrations. A second objective is the classification of higher Fano manifolds, the study of their connection to higher rational connectedness, and the study of their complex differential geometry. The final objective is to study the parameter spaces for rational curves on Fano manifolds which do not satisfy the higher Fano conditions. This is relevant to the open problem of proving existence of non-unirational Fano manifolds.Quite frequently in science and engineering one wants to solve a system of polynomial equations in some set of variables, and depending on some set of parameters. An optimal case is when there is a solution to the system of equations whose coordinates are themselves polynomials (or more often fractions of polynomials) in the parameters. There are some sufficient conditions for this optimal solution which involve the geometry of the solution set for a general choice of the parameters. The goal is to sharpen these results to give conditions which are both sufficient and necessary, i.e., to develop a theory of "obstructions" to the existence of rational solutions.The educational component of the proposal has 3 parts: a program aimed at training high school math teachers from the MA program directed by the PI so that they may establish and run math clubs and math circles in their schools, a seminar/summer workshop in mathematical exposition for graduate students and recent postdocs, and a northeastern regional algebraic geometry seminar fostering interactions between graduate students in different cities across the northeastern United States.