Complex Dynamics and Moduli Spaces

复杂动力学和模空间

• 批准号: 1305116
• 负责人: Curtis McMullen
• 金额: $ 44.64万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1305116&HistoricalAwards=false
• 关键词: Complex; Dynamics; Moduli; Spaces

Moduli spaces of lattices, Riemann surfaces, rational maps and other algebraic structures exhibit rich geometry, often accompanied by rigidity and a connection with arithmetic. These spaces also have a dynamical nature -- they support natural flows or group actions with complicated orbits, or they classify such actions. This projects aims to investigate moduli spaces from a dynamical perspective. In the setting of Riemann surfaces we study Teichmueller curves, Jacobians with real multiplication, volumes of complex hyperbolic cone manifolds and representations of the braid group developed by Picard, Deligne-Mostow and Thurston. We also investigate dynamical systems of low entropy, ranging from pseudo-Anosov maps to automorphism of K3 surfaces, via constructions adapted to small Salem numbers and Coxeter groups. In the setting of lattices we aim to investigate packing constants of ideals in number fields, and to demonstrate an abundance of bounded orbits for rank one flows, to complement the dearth of such orbits predicted in higher rank by conjectures of Littlewood and Margulis. Finally, in the setting of iterated rational maps in one complex variable, we study the natural bifurcation measure on moduli spaces attached to individual critical points, and aim to show their independence and transversality, with an eye towards arithmetic applications. We also intend to link the theory of Berkovitch spaces to the degeneration of hyperbolic manifolds to trees.From particle physics to finance, from evolution to climate change, the world is full of dynamical systems. Simple algebraic transformations already exhibit many of the features of these natural phenomena, such as phase transitions and tipping points that signal the onset of new regimes. These universal patterns may be revealed through the rigorous study of moduli spaces, their compactifications and their stratifications by dynamical invariants. This project appeals to a broad range of mathematical disciplines to both deepen our understanding of dynamical systems and to sharpen our mathematical and computational methods.

格、黎曼曲面、有理映射和其他代数结构的模空间表现出丰富的几何形状，通常伴随着刚性和与算术的联系。这些空间也具有动态性质——它们支持具有复杂轨道的自然流动或群体行动，或者它们对这些行动进行分类。本项目旨在从动力学角度研究模空间。在黎曼曲面的背景下，我们研究了Teichmueller曲线、带实乘法的雅可比矩阵、复双曲锥流形的体积以及由Picard、delign - mostow和Thurston提出的辫群的表示。我们也研究了低熵的动力系统，范围从伪anosov映射到K3曲面的自同构，通过构造适应于小Salem数和Coxeter群。在格的设置下，我们的目的是研究理想数域的填充常数，并证明秩一流的有界轨道的丰富性，以补充Littlewood和Margulis猜想在高秩流中预测的有界轨道的缺乏。最后，在一个复变量的迭代有理映射的情况下，我们研究了附加于单个临界点的模空间上的自然分岔测度，旨在证明它们的独立性和横切性，并着眼于算术应用。我们也打算将Berkovitch空间理论与双曲流形退化为树联系起来。从粒子物理到金融，从进化到气候变化，这个世界充满了动力系统。简单的代数变换已经表现出这些自然现象的许多特征，例如标志着新政权开始的相变和临界点。这些普遍模式可以通过对模空间、模空间的紧化和模空间的动态不变量分层的严格研究来揭示。这个项目吸引了广泛的数学学科，既加深了我们对动力系统的理解，又提高了我们的数学和计算方法。