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具有丰富的几何结构，往往伴随着刚性和与算术的联系。这些空间还具有动力学性质--它们支持自然流动或具有复杂轨道的群体动作，或者它们对此类动作进行分类。本项目旨在从动力学的角度研究模空间。在Riemann曲面的背景下，我们研究了Teichmueller曲线、实乘雅可比、复双曲锥流形的体积以及Picard、Deligne-Mostow和瑟斯顿发展的辫子群的表示。我们还通过适合于小Salem数和Coxeter群的构造，研究了从伪Anosov映射到K3曲面的自同构的低熵动力系统。在格的背景下，我们的目的是研究数域中理想的填充常数，并证明一阶流动的大量有界轨道，以补充Littlewood和Marguis猜想在较高阶预测的轨道的不足。最后，在单复变量迭代有理映射的背景下，我们研究了依附于单个临界点的模空间上的自然分歧度量，目的在于证明它们的独立性和横截性，并着眼于算术应用。我们还打算将Berkovitch空间的理论与双曲流形到树的退化联系起来。从粒子物理到金融，从进化到气候变化，世界上充满了动力系统。简单的代数变换已经表现出这些自然现象的许多特征，例如相变和预示新制度开始的临界点。这些普适模式可以通过严格研究模空间及其紧致化和动力学不变量的层次化来揭示。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，也加强了我们的数学和计算方法。