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

有理连通性、有理简单连通性等是路径连通性、简单连通性等的代数几何类比。正如高连通性和高同伦群在拓扑阻塞理论中占有重要地位一样，代数-几何类似物也应该在代数-几何阻塞理论中占有重要地位。特别是，来自a1 -同伦理论的新思想将解决Hassett和Tschinkel的弱逼近问题。研究部分有三个目标。一个目标是发展类似于拓扑阻塞理论的高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截面的存在性。第二个目标是高范诺流形的分类，研究它们与高有理连通性的联系，以及研究它们的复杂微分几何。最后的目的是研究不满足高范诺条件的范诺流形上有理曲线的参数空间。这与证明非酉范诺流形存在性的开放问题有关。在科学和工程中，人们经常想要解一个多项式方程组在一些变量集合中，依赖于一些参数集合。最优的情况是当方程组的解的坐标本身是参数中的多项式（或更经常是多项式的分数）时。这种最优解存在一些充分条件，这些条件涉及解集的几何形状，用于参数的一般选择。我们的目标是使这些结果更加尖锐，从而给出充分和必要的条件，也就是说，发展一种理性解存在的“障碍”理论。该提案的教育部分有三个部分：一个旨在培训由PI指导的硕士课程的高中数学教师，使他们能够在他们的学校建立和经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研讨会/夏季研讨会；以及一个东北地区代数几何研讨会，促进美国东北部不同城市研究生之间的互动。

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.