Higher rational connectedness and applications

有理单连通性是一个代数概念，它与单连通性的关系就像有理连通性与路连通性的关系一样，就像二维基上具有单连通纤维的拓扑纤维化有一个连续截面一样，曲面上具有有理单连通一般纤维的代数纤维化也有一个有理截面（在适当的附加假设下）。 本课题研究的是超越这个结果的理论，正如拓扑阻塞理论是超越所引用的拓扑结果的理论一样。第一个目标是确定“弱近似”（多项式近似幂级数解）的障碍如何分解为局部障碍和全局障碍。 第二个目标是研究障碍理论，它还不知道通过精确地确定哪些代数纤维化在一个指定的，简单的类型的表面承认合理的部分。系统的多项式方程是无处不在的数学，科学和工程。 在研究复数中所有解的集合时，即，与这样的系统相关联的多样性，有一个特殊的现象：如果对于每一对解，有一个多项式映射在多样性中取值，并且其值在给定的一对解之间插值，则系统是“理性连通的”。 这种特殊性质在实践中经常得到满足。 令人惊讶的是，一个多项式方程组在代数上依赖于一个额外的参数（通常被认为是时间）总是有一个家庭的解决方案作为一个多项式的参数，只要系统的一个固定的一般选择的参数是合理的连接。 现在有一个2参数系统的类似定理，但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中的概念，即，“橡胶板几何”。

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".