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

有理连通性、有理单连通性等是路连通性、单连通性等的代数几何模拟，正如高连通性和高同伦群在拓扑阻塞理论中起着重要作用一样，代数几何类似在代数几何阻塞理论中也应该发挥重要作用。特别是，来自A1-同伦理论的新思想应该解决Hassett和Tschinkel的弱逼近问题。研究部分有三个目标。一个目标是发展一个类似于拓扑阻塞理论的高次有理连通性的代数几何理论，并将其应用于代数纤维的有理部分的存在。第二个目标是对高阶Fano流形进行分类，研究它们与高阶有理连通性的联系，并研究它们的复微分几何。最后研究了Fano流形上不满足高阶Fano条件的有理曲线的参数空间。这与证明非么正Fano流形的存在性这一公开问题有关。在科学和工程中，人们经常想要解一个多项式方程组，它包含某组变量，并且依赖于某组参数。最好的情况是，当方程组有解时，其坐标本身就是参数中的多项式(或者更常见的是多项式的分数)。对于参数的一般选择，存在与解集几何有关的最优解的充分条件。该提案的教育部分包括三个部分：一个旨在培训高中数学教师的计划，以便他们可以在自己的学校建立和运营数学俱乐部和数学圈；一个为研究生和最近的博士后举办的数学博览会研讨会/暑期研讨会；以及一个促进美国东北部不同城市研究生之间互动的东北地区代数几何研讨会。

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.