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Dynamics and Hodge theory: Uniformization and Bialgebraic Geometry

动力学和霍奇理论：均匀化和双代数几何

基本信息

批准号：
2305394
负责人：
Simion Filip
金额：
$ 37万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305394&HistoricalAwards=false
关键词：
Dynamics Hodge theory Uniformization Bialgebraic

项目摘要

Research in dynamical systems aims to understand the long-term behavior of a structure that changes according to some prescribed law. The structure and the law can come from diverse fields such as physics, economics, biology, etc. and as a consequence dynamical systems pervade most areas of science and its applications. The PI will study a broad class of low-dimensional dynamical systems using tools from other areas of mathematics, notably algebraic geometry and Hodge theory. The tools and techniques that will be developed will cross-fertilize these mathematical disciplines, addressing longstanding problems and opening new directions of investigation. The research program will also be suitable for engaging and training early career mathematicians, including graduate students and postdocs.In one direction, the PI will study the relationship between Hodge theory and a class of dynamical systems called Anosov representations. These connections were only recently unraveled by the PI and promise to enrich both fields. In particular, longstanding obstacles related to the mysterious differential geometry of period maps in Hodge theory can be overcome by considering other target spaces in flag manifolds, suggested by dynamics, and which allow for uniformization results of a new kind. In a second direction, the PI will study dynamics in moduli spaces of translation surfaces with new tools coming from o-minimality and transcendence theory. These tools will be used to shed light on classification problems that are central to the subject. In addition, existing finiteness results will be made effective and algorithms will be devised to efficiently compute orbit closures.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.

动力系统研究的目的是了解一个结构的长期行为，根据一些规定的法律变化。结构和规律可以来自不同的领域，如物理学，经济学，生物学等，因此动力系统渗透到大多数科学领域及其应用。PI将使用其他数学领域的工具研究广泛的低维动力系统，特别是代数几何和霍奇理论。将开发的工具和技术将交叉施肥这些数学学科，解决长期存在的问题，并开辟新的研究方向。该研究计划也将适合从事和培训早期职业数学家，包括研究生和博士后。在一个方向上，PI将研究霍奇理论和一类称为Anosov表示的动力系统之间的关系。这些联系直到最近才被PI解开，并有望丰富这两个领域。特别是，长期存在的障碍有关的神秘微分几何的周期映射霍奇理论可以克服考虑其他目标空间的旗流形，建议的动力学，并允许一种新的均匀化结果。在第二个方向，PI将研究动态模空间的平移表面与新的工具来自o-极小和超越理论。这些工具将被用来阐明分类问题，这是核心的主题。此外，现有的有限性结果将变得有效，并将设计算法来有效地计算轨道关闭。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。