CAREER: Interpolation, stability, and rationality

这个项目开发的技术，以了解几何的解决方案，以系统的多项式方程。 特别感兴趣的是，当一组解决方案是一维的，在这种情况下，它被称为代数曲线。 由于多项式在科学和工程中无处不在，这样的解集出现在许多不同的上下文中。 一个重要的例子是插值问题：给定一个一般点的集合，什么时候有一个固定类型的代数曲线通过这些点？ 这个问题在密码学和信息论中都有应用。 这些研究项目将揭示多项式方程的代数曲线的可能实现，以及插值问题的新的和重要的情况。 这将辅之以教育和推广活动，包括为早期到中期职业妇女和代数几何和指导本科生研究的非二元研究生设计的一系列讲习班。更具体地说，研究项目在以下三个方向。 第一个集中在自然分层的空间向量丛的曲线上配备了一个固定的优势映射到另一条曲线的稳定性的推进。 第二个集中在其他设置中的插值问题，包括当环境变量是齐次空间时。 最后一个重点是合理性问题的品种在非封闭的地面fields，特别是在概括的中间雅可比torsor obligation.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This project develops techniques to understand the geometry of the solutions to systems of polynomial equations. Of particular interest is when the set of solutions are one-dimensional, in which case it is called an algebraic curve. Since polynomials are ubiquitous in science and engineering, such solution sets arise in many different contexts. One important example is the interpolation problem: given a collection of general points, when is there a fixed type of algebraic curve passing through these points? This problem has applications to cryptography and information theory. The research projects will shed light on the possible realizations of an algebraic curve by polynomial equations, as well as new and important cases of the interpolation problem. This will be complemented by educational and outreach activities, including a sequence of workshops designed for early-to-mid career women and nonbinary graduate students in algebraic geometry and mentoring undergraduate research.More specifically, the research projects are in the following three directions. The first focuses on the natural stratification of the space of vector bundles on a curve equipped with a fixed dominant map to another curve by the stability of the pushforward. The second focuses on the interpolation problem in other settings, including when the ambient variety is a homogeneous space. The last focuses on the rationality problem for varieties over nonclosed ground fields, and in particular upon generalizations of the intermediate Jacobian torsor obstruction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.