Geometric counts on surfaces

曲面是到处都是二维的几何形状。这样的几何对象在纯数学以及科学和工程的各个领域中起着重要的作用。这个项目的重点是理解度量和表面的几何属性，即长度和角度的概念。曲面的度量结构变形产生所谓的“模数空间”;模数空间中的每一点代表底层曲面上的一个特定几何形状。PI将继续研究表面几何的两个自然类别：（i）奇异平坦度量和（ii）双曲（负弯曲）度量。对平坦度量的进一步理解已经并将继续导致动力系统的新结果。从某种意义上说，双曲线度量是放在这些曲面上最自然的度量。本项目还将研究曲面模空间上的各种结构，特别是测度，它提供了一个定义明确的“随机曲面”概念。PI将使用并进一步开发两种度量之间的类比，以及与图论，概率论，谱几何和代数几何的联系。 该项目的更广泛的影响包括生成适合研究生的问题和主题。PI将继续参与旨在将早期职业研究人员聚集在一起的计划，集中和协作的设置工作的问题。在平坦的度量设置，PI使用了新的多尺度紧化层的翻译表面证明强正则性遍历SL_2不变的概率措施，验证了自然启发式仿射不变流形。这一结果及其证明中的技巧已经被其他人应用于研究随机Teichmuller测地线和鞍联络对的计数。在另一项联合工作中，PI证明了可以在多尺度边界附近定义仿射不变流形的方程类型的限制，从而提供了赖特圆柱变形定理的新证明，并将其推广到亚纯地层。PI将使用上述结果和技术通过退化来研究仿射不变流形的重要分类问题，以及关于较少研究的k2的k-微分的问题。PI还开发了一个关于随机双曲曲面的研究计划，这是一个年轻而活跃的领域。与萨皮尔一起，PI已经证明了一个关于简单与非简单闭测地线的相对数量的猜想，对于各种随机曲面的概念，因为亏格趋于无穷大。 在随机双曲曲面和随机正则图之间有一个富有成效的类比; PI已经解决了硬币两面的问题，并将继续研究。 这项研究导致进一步的问题，一个典型的测地线的形状在一个通用的表面，以及某些问题的测地线在每一个双曲曲面。这个奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Surfaces are geometric shapes that are everywhere two dimensional. Such geometric objects play an important role in pure mathematics, as well as in various areas of science and engineering. This project focuses on understanding metric and geometric properties of surfaces, i.e. notions of length and angle. Deforming the metric structure of a surface gives rise to so-called `moduli spaces’; each point in the moduli space represents a particular geometry on the underlying surface. The PI will continue to investigate two natural classes of surface geometry: (i) singular flat metrics, and (ii) hyperbolic (negatively curved) metrics. Improved understanding of flat metrics has led, and will continue to lead, to new results in dynamical systems. Hyperbolic metrics are, in a certain sense, the most natural metrics to put on these surfaces. This project will also investigate various structures on the moduli spaces of surfaces, notably measures, which provide for a well-defined notion of `random surface’. The PI will use and further develop analogies between the two types of metrics, as well as connections to graph theory, probability theory, spectral geometry, and algebraic geometry. The broader impact of this project includes the generation of questions and topics suitable for graduate students. The PI will maintain an involvement with programs aimed at bringing together early-career researchers to work on problems in concentrated and collaborative settings.In the flat metric setting, the PI has used the new multi-scale compactification of strata of translation surfaces to prove strong regularity of ergodic SL_2-invariant probability measures, verifying a natural heuristic about affine invariant manifolds. This result, and the techniques in its proof, have already been applied by others to study random Teichmuller geodesics and counts of pairs of saddle connections. In another joint work, the PI has proved restrictions on the type of equations that can define an affine invariant manifold near the multi-scale boundary, thereby providing a new proof of Wright's Cylinder Deformation Theorem and generalizing it to meromorphic strata. The PI will use the above results and techniques to study, via degeneration, the centrally important classification problem for affine invariant manifolds, as well as problems about the less studied k-differentials for k2. The PI has also developed a research program concerning random hyperbolic surfaces, a young and active field. Together with Sapir, the PI has proved a conjecture concerning the relative number of simple versus non-simple closed geodesics for various notions of random surfaces, as genus tends to infinity. There is a fruitful analogy between random hyperbolic surfaces and random regular graphs; the PI has solved, and continues to investigate, problems on both sides of the coin. This research leads to further questions about the shape of a typical geodesic on a generic surface, as well as to certain questions about geodesics on every hyperbolic surface.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.