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CAREER: Birational Geometry and Algebraic Dynamics

职业：双有理几何和代数动力学

基本信息

批准号：
2142966
负责人：
John Lesieutre
金额：
$ 45万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-05-01 至 2027-04-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142966&HistoricalAwards=false
关键词：
CAREER Birational Geometry Algebraic Dynamics

项目摘要

This project focuses on the role of symmetry in algebraic geometry, which is the study of the solutions of systems of polynomial equations in many variables, and methods of exploiting symmetries in these systems to better understand their structure. Solving such systems is often very difficult: in general, it is hard to learn much about the set of solutions unless one knows some peculiarities of the specific equations at hand. This research lies at the intersection of algebraic geometry and dynamical systems, which studies the behavior of repeated applications of functions on certain sets of points. A key component of this project is to bring the techniques from each of these disciplines to bear on questions from the other. The project includes educational components at every level. At the most advanced levels, the project focuses on introducing algebraic dynamics to new audiences, including graduate students in dynamics and researchers in algebraic geometry. The project will also involve several undergraduate researchers learning the basics of the field, as well as the development of educational materials for younger students hoping to understand the breadth of mathematics beyond calculus.More technically, a primary goal of the project is to understand the growth of degrees of iterates of rational maps, from both computational and theoretical points of view. There has been considerable effort directed at understanding lower bounds on degree growth, and the investigators will use new methods from the Sarkisov program to study this question. The researchers will also investigate computational questions and improve the capacity of the open-source SageMath system to compute dynamical degrees. In other directions, the researchers will focus on questions related to the growth rates of spaces of sections of tensor powers of line bundles as well as higher-dimensional versions of the bounded negativity conjecture.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.

该项目的重点是对称性在代数几何中的作用，这是研究多变量多项式方程组的解决方案，以及利用这些系统中的对称性以更好地理解其结构的方法。求解这样的方程组通常是非常困难的：一般来说，除非知道手头的特定方程的一些特性，否则很难了解解的集合。这项研究是在代数几何和动力系统的交叉点，它研究的行为重复应用的功能，在某些点集。该项目的一个关键组成部分是将每个学科的技术应用于另一个学科的问题。该项目包括各级教育部分。在最先进的水平，该项目的重点是介绍代数动力学的新观众，包括研究生在动力学和研究人员在代数几何。该项目还将涉及一些本科生研究人员学习该领域的基础知识，以及为希望了解微积分之外的数学广度的年轻学生开发教育材料。更技术上，该项目的主要目标是从计算和理论的角度理解有理映射迭代次数的增长。在理解学位增长的下限方面已经做出了相当大的努力，研究人员将使用Sarkisov计划的新方法来研究这个问题。研究人员还将研究计算问题，并提高开源SageMath系统计算动态度的能力。在其他方向，研究人员将专注于与线丛张量幂部分的空间增长率以及有界负性猜想的高维版本相关的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。