CAREER: Birational Geometry and Algebraic Dynamics

这个项目集中于对称性在代数几何中的作用，它是研究多变量多项式方程组的解，以及利用这些系统中的对称性来更好地了解它们的结构的方法。求解这类系统通常是非常困难的：一般来说，除非知道手头特定方程的一些特性，否则很难对这组解有太多了解。这项研究位于代数几何和动力系统的交叉点上，研究函数在某些点集上重复应用的行为。这个项目的一个关键组成部分是将这些学科的技术与其他学科的问题联系起来。该项目包括各个级别的教育内容。在最高级的水平上，该项目专注于向新的受众介绍代数动力学，包括动力学研究生和代数几何研究人员。该项目还将包括几名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以外的数学广度的年轻学生开发教材。更严格地说，该项目的一个主要目标是从计算和理论的角度了解有理映射迭代次数的增长。为了了解学位增长的下限，已经做出了相当大的努力，研究人员将使用萨基索夫项目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系统计算动力学度数的能力。在其他方面，研究人员将专注于与线丛张量幂的部分空间的增长率以及有界负性猜想的高维版本相关的问题。这一奖项反映了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 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.