CAREER: Interpolation, stability, and rationality

这个项目发展了理解多项式方程组解的几何的技术。特别有趣的是当解的集合是一维的，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环境中。一个重要的例子是插值问题：给定一个一般点的集合，什么时候会有一条固定类型的代数曲线经过这些点？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，以及插值问题的新的和重要的情况。这将辅以教育和外联活动，包括为早期到中期职业妇女和代数几何非二进制研究生设计的一系列讲习班，并指导本科生的研究。更具体地说，研究项目在以下三个方向。第一个重点是在一条曲线上的向量束空间的自然分层，这条曲线配备了一个固定的优势映射到另一条曲线，这是通过向前推进的稳定性实现的。第二个重点是在其他设置下的插值问题，包括当环境变量是齐次空间时。最后，重点讨论了非闭场上变量的合理性问题，特别是对中间雅可比扭转阻力的推广。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.