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| Complex Dynamics and Moduli Spaces | |
| 项目编号 | 1903764 |
| Curtis McMullen | |
| 项目主持机构 | Harvard University |
| 开始日期 | 2019-07-01 |
| 结束日期 | 06/30/2024 |
| 英文摘要 | 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. Its methods have already led to the discover of new and unexpected algebraic regimes, through a combination of theoretical tools that narrow the domain of search, and experimental methods such as the simulation of bouncing molecules in idealized chambers. A central issue in science, from biology to mathematics, is the study and classification of the wide variations that can take place in a single recognized species. The moduli spaces constructed by Ahlfors, Bers and Mumford in the 1960s, give this kind of classification for compact Riemann surfaces with a fixed genus g and number of marked points n. Today the study of moduli spaces is the meeting ground for disciplines ranging from arithmetic geometry to string theory. In dimension three, Riemann surfaces are replaced by hyperbolic 3-manifolds. This project aims to expand the frontiers of our understanding of both moduli spaces and hyerbolic 3-manifolds from the perspective of dynamical systems, unified by the action of SL2(R) in both regimes. It draws on methods ranging from complex analysis to low--dimensional topology and renormalization. The concerted study of specific examples, assisted by computer visualization and algebraic manipulation, also plays a central role of in this research. The PI and his coworkers have recently discovered new totally geodesic curves and surfaces in the moduli spaces for (g,n) = (4,0), (1,3), (1,4) and (2,1). Their methods also reproduce most previously known examples of these rare and beautiful objects. A central goal of this project is to put forth a unified construction of all known examples, and investigate the prospects for obtaining a complete classification. The project will also address problems concerning totally geodesic planes in open hyperbolic 3-manifolds, the complexity of closed loops on surfaces, the arithmetic underlying billiards in polygons, moduli spaces of rational maps, and connection between twisted 1-forms and 3-manifolds that fiber over the circle. The training of graduate students to become active and independent researchers forms a central part of this project. The field of complex dynamics and moduli spaces is one where concrete problems and examples abound, and yet some of the deepest insights and methods of modern mathematics can be brought to bear. It invites the participation of experts from a broad range of fields. This project will continue to foster the growth of an open, diverse and active research network. 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. |
| 资助机构 | US-NSF |
| 项目经费 | $600,000.00 |
| 项目类型 | Continuing Grant |
| 国家 | US |
| 语种 | 英语 |
| 文献类型 | 项目 |
| 条目标识符 | http://gcip.llas.ac.cn/handle/2XKMVOVA/212344 |
| 推荐引用方式 GB/T 7714 | Curtis McMullen.Complex Dynamics and Moduli Spaces.2019. |
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