Higher rational connectedness and applications

有理单连通性是一个代数概念，它对简单连通性的作用就像有理连通性对路径连通性的作用一样，就像具有单连通纤维的二维基上的拓扑纤维允许连续截面一样，具有有理单连通普通纤维的曲面上的代数纤维也允许有理截面(在适当的附加假设下)。本课题对这一结果之外的理论进行了研究，就像拓扑阻塞理论是引用的拓扑结果之外的理论一样。第一个目标是确定对“弱逼近”(用多项式解逼近幂级数解)的障碍如何分解为局部障碍和整体障碍。第二个目标是研究阻塞理论，在这个理论中，通过精确地确定指定的简单类型的曲面上的哪些代数纤维允许有理截面，来研究该理论。多项式方程组在数学、科学和工程中普遍存在。在研究与这样一个系统相关的复数中的所有解的集合，即簇，有一个特殊的现象：如果对于每一对解，存在一个取值于簇中的值的多项式映射，并且其值在给定的解对之间内插，则系统是“有理连通的”。这一特殊性质往往在实践中得到满足。令人惊讶的是，代数上依赖于一个额外参数(通常被认为是时间)的多项式方程系统总是有一族解作为参数的多项式变化，只要系统对于固定的一般参数选择是有理连接的。现在有一个关于两参数系统的类似定理，但对系统有非常强的约束。该项目的目标是通过使用与拓扑学中的概念类似的概念，即“橡胶片几何”，来削弱约束条件，从而使推进更广泛地适用。

Rational simple connectedness is an algebraic notion which is to simple connectedness as rational connectedness is to path connectedness.Just as a topological fibration over a 2-dimensional base with simply connected fiber admits a continuous section, also an algebraic fibration over a surface with rationally simply connected general fiber admits a rational section (under suitable additional hypotheses). This project investigates the theory beyond this result, just as topological obstruction theory is the theory beyond the quoted topological result. The first goal is to determine how the obstruction to "weak approximation"(approximation of power series solutions by polynomial solutions) decomposes into a local obstruction and a global obstruction. The second goal is to investigate the obstruction theory where it is not yet known by determining precisely which algebraic fibrations over a surface of a specified, simple type admit a rational section.Systems of polynomial equations are ubiquitous in mathematics, science and engineering. In studying the collection of all solutions in complex numbers, i.e., the variety, associated to such a system, there is one special phenomenon: the system is "rationally connected" if for every pair of solutions, there is a polynomial map taking values in the variety and whose values interpolate between the given pair of solutions. This special property is often satisfied in practice. Surprisingly, a system of polynomial equations depending algebraically on 1 extra parameter (often thought of as time) always has a family of solutions varying as a polynomial of the parameter so long as the system for a fixed general choice of the parameter is rationally connected. There is now an analogous theorem for a 2-parameter system, but with very strong constraints on the system. The goal of the project is to weaken the constraint condition, and thus make the advance more widely applicable, by using notions analogous to those in topology, i.e., "rubber-sheet geometry".