Dynamics and Hodge theory: Uniformization and Bialgebraic Geometry

动力系统研究的目的是了解一个结构的长期行为，该结构根据某些规定的规律变化。其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科学及其应用的大多数领域。PI将使用其他数学领域的工具，特别是代数几何和霍奇理论，研究一类广泛的低维动力系统。将开发的工具和技术将使这些数学学科交叉发展，解决长期存在的问题并开辟新的研究方向。该研究项目也将适合从事和培养早期职业数学家，包括研究生和博士后。在一个方向上，PI将研究Hodge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揭开，并有望丰富这两个领域。特别是，与Hodge理论中周期图的神秘微分几何相关的长期障碍可以通过考虑动力学建议的旗流形中的其他目标空间来克服，并且允许一种新的均匀化结果。在第二个方向上，PI将使用来自极小性和超越理论的新工具研究平移曲面的模空间中的动力学。这些工具将用于阐明本主题的核心分类问题。此外，将使现有的有限性结果有效，并设计算法以有效地计算轨道闭合。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.