权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Birational Geometry and Algebraic Dynamics

双有理几何和代数动力学

基本信息

批准号：
1700898
负责人：
John Lesieutre
金额：
$ 13.27万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2019-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700898&HistoricalAwards=false
关键词：
Birational Geometry Algebraic Dynamics

项目摘要

This project focuses on the role of symmetry in algebraic geometry, an area of mathematics concerned with the study of the solution sets of systems of polynomial equations. This topic has long played a central role in mathematics, and it lies at the intersection of many fields of mathematics: the problems to be studied relate to wide variety of areas of mathematics, including topology, algebra, and number theory. The project will investigate a variety of situations in which unexpected symmetries of equations make it possible to explore otherwise difficult-to-understand geometric phenomena and to test a wide variety of geometric conjectures in new settings.The particular emphasis of this project will be on the connections between algebraic geometry and algebraic dynamics, the study of iterated rational maps, and the results will include applications of each field to the other. On one hand, the investigator will work to understand how ideas from dynamics can be used to reveal a variety of otherwise obscure phenomena in higher-dimensional algebraic geometry, and he aims to demonstrate that varieties admitting dynamically interesting self-maps provide a fertile source of examples (and counterexamples). Conversely, the investigator will also work to understand how the methods of algebraic geometry (and the minimal model program in particular) can be employed to understand the geometric properties of algebraic varieties with very large groups of symmetries.

该项目重点关注对称性在代数几何中的作用，代数几何是一个涉及多项式方程组解集研究的数学领域。该主题长期以来在数学中发挥着核心作用，并且它位于许多数学领域的交叉点：要研究的问题涉及数学的各个领域，包括拓扑、代数和数论。该项目将研究各种情况，在这些情况下，方程的意外对称性使得探索原本难以理解的几何现象成为可能，并在新的环境中测试各种几何猜想。该项目的特别重点将是代数几何和代数动力学之间的联系，迭代有理图的研究，其结果将包括每个领域到另一个领域的应用。一方面，研究者将努力理解如何利用动力学的思想来揭示高维代数几何中各种原本晦涩的现象，他的目标是证明承认动态有趣的自映射的品种提供了丰富的例子（和反例）来源。相反，研究者还将努力了解如何使用代数几何方法（特别是最小模型程序）来理解具有非常大的对称群的代数簇的几何性质。