Higher rational connectedness and applications

理性简单连通性是一个代数概念，它之于简单连通性就像理性连通性之于路径连通性一样。正如在单连通纤维的二维基底上的拓扑纤维允许一个连续截面一样，在具有合理单连通一般纤维的表面上的代数纤维也允许一个合理截面（在适当的附加假设下）。本课题研究的是这个结果之外的理论，正如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”（幂级数解的多项式逼近）的阻碍如何分解为局部阻碍和全局阻碍。第二个目标是通过精确地确定在特定的简单类型表面上的哪些代数纤维允许有理截面来研究尚不知道的阻碍理论。多项式方程组在数学、科学和工程中无处不在。在研究与这类系统相关的所有复数解的集合即变种时，有一个特殊的现象：如果对于每一对解，在变种中存在一个取值的多项式映射，且其值在给定的对解之间插入，则该系统是“合理连通的”。这种特殊性质在实践中经常得到满足。令人惊讶的是，一个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个随参数多项式变化的解族，只要这个系统有一个固定的一般参数选择是合理连接的。现在有一个2参数系统的类似定理，但是对系统有很强的约束。该项目的目标是通过使用类似于拓扑学的概念，即“橡胶板几何”，削弱约束条件，从而使进展更广泛地适用。

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".