The SL(2,R) action on moduli space

模空间上的 SL(2,R) 作用

• 批准号: 1201422
• 负责人: Alex Eskin
• 金额: $ 31.4万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201422&HistoricalAwards=false
• 关键词: SL; action; moduli; space

The proposal concerns the interrelated analytic study of billiards in rational polygons, moduli spaces of abelian and quadratic differentials, and the dynamics of the action by the group of two-by-two matrices on these moduli spaces. In recent work with M.Mirzakhani, the PI was able to prove some dynamical rigidity results for this action, which allow one to understand every (and not just almost every) orbit. This is important for several reasons. In particular, the surfaces which arise from billiards are a set of measure zero in the moduli space, and ergodic theorems which hold at every point are needed to prove results about billiards. Many of the results and techniques are based on a loose analogy with the theory of unipotent flows on locally symmetric spaces (e.g. Ratner's theorem). However, the moduli spaces of differentials are substantially different, and new ideas were needed. The PI proposes additional research in this direction.Some natural phenomena are ``chaotic'' (i.e. unpredictable). These are often studied by statistical methods. Others are ``integrable'' (i.e. predictable and regular). Other phenomena fit somewhere in between. The polygonal billiard system, which is one of the main subjects of study of the proposal, is a good model of intermediate behavior. As such it has been studied extensively in physics as well, in particular in connection to``quantum chaos''. The PI believes that the new techniques and tools introduced will have applications in these and other fields.

该建议涉及相关的分析研究台球在合理的多边形，模空间的阿贝尔和二次微分，和动态的行动组的两个由两个矩阵对这些模空间。 在最近与M.Mirzakhani的合作中，PI能够证明这种作用的一些动力学刚性结果，这使得人们能够理解每一个（而不仅仅是几乎每一个）轨道。这一点很重要，原因有几个。特别是，表面所产生的台球是一组措施零的模空间，遍历定理，在每一点都需要证明结果有关台球。许多结果和技巧都是基于与局部对称空间上的单幂流理论（例如拉特纳定理）的松散类比。 然而，微分的模空间有很大的不同，需要新的想法。 PI建议在这个方向上进行更多的研究。有些自然现象是“混乱的”（即不可预测的）。这些通常是通过统计方法来研究的。另一些是“可整合的”（即可预测和定期的）。其他现象介于两者之间。多边形台球系统，这是该建议的主要研究课题之一，是一个很好的模型的中间行为。因此，它在物理学中也得到了广泛的研究，特别是与“量子混沌”有关。PI相信，新技术和新工具将在这些领域和其他领域得到应用。