CAREER: Interpolation, stability, and rationality

这个项目开发了理解多项式方程组的解的几何形状的技术。特别令人感兴趣的是当解的集合是一维的时候，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背景下。一个重要的例子是插补问题：给定一个概念点的集合，何时会有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。这些研究项目将阐明用多项式方程实现代数曲线的可能性，以及插补问题的新的和重要的情况。此外，还将开展教育和外展活动，包括为职业生涯早期到中期的女性和非二进制研究生举办一系列代数几何和指导本科生研究的研讨会。第一个集中讨论了一条曲线上的向量丛空间的自然分层以及一条固定的占优映射对另一条曲线的稳定性的推进。第二个重点是在其他环境下的内插问题，包括当环境变化是均匀空间时。最后一项集中在非封闭地面上的品种的合理性问题，特别是对中间雅可比扭矩障碍的一般化。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.