CAREER: Interpolation, stability, and rationality

职业：插值、稳定、合理

• 批准号: 2338345
• 负责人: Isabel Vogt
• 金额: $ 54.95万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2029-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2338345&HistoricalAwards=false
• 关键词: CAREER; Interpolation; stability; rationality

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.

该项目开发了了解多项式方程系统解决方案的几何形状的技术。 特别有趣的是，当解决方案集为一维时，在这种情况下，它称为代数曲线。 由于多项式在科学和工程中无处不在，因此这种解决方案集在许多不同的情况下出现。 一个重要的例子是插值问题：给定一般点的集合，何时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理论上有应用。 研究项目将阐明多项式方程式代数曲线的可能实现，以及插值问题的新和重要情况。 这将由教育和外展活动进行补充，包括一系列专为代数几何学和指导本科研究的早期至中等职业女性和非培养研究生设计的研讨会。更具体地说，研究项目在以下三个方向上。 第一个侧重于通过推动力的稳定性，在配备有固定优势地图的曲线上的矢量束空间的自然分层。 第二个重点是在其他设置中的插值问题，包括何时环境品种是一个均匀的空间。 最后的重点是在未公开的地面上的品种的合理性问题，尤其是在中间雅各布·托索尔阻塞的概括上。这项奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的审查标准来通过评估来通过评估来支持的。