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.

格的模空间、黎曼曲面、有理映射和其他代数结构表现出丰富的几何学，通常伴随着刚性和与算术的联系。这些空间也有一个动态的性质-它们支持自然流或具有复杂轨道的群体行为，或者它们对这些行为进行分类。 这个项目旨在从动力学的角度研究模空间。 在设置的黎曼曲面，我们研究Teichmueller曲线，雅可比与真实的乘法，体积的复杂双曲锥流形和表示的辫子群开发的皮卡德，Deligne-Mostow和瑟斯顿。 我们还研究动力系统的低熵，从伪Anosov映射的K3表面的自同构，通过适应小塞勒姆数和Coxeter群的建设。 在设置格，我们的目标是调查包装常数的理想在数域中，并证明一个丰富的有界轨道的秩一流，补充缺乏这样的轨道预测在更高的排名由prostitutures的Littlewood和马古利斯。最后，在一个复变的迭代有理映射的设置下，我们研究了附加到各个临界点的模空间上的自然分歧测度，旨在表明它们的独立性和横截性，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，世界充满了动力系统。 简单的代数变换已经展示了这些自然现象的许多特征，例如相变和临界点，它们标志着新制度的开始。这些普遍的模式可以通过模空间的严格研究，它们的紧化和动力学不变量的分层来揭示。 这个项目呼吁广泛的数学学科，既加深我们对动力系统的理解，又提高我们的数学和计算方法。