Geometric counts on surfaces

表面上的几何计数

• 批准号: 2247244
• 负责人: Benjamin Dozier
• 金额: $ 33.06万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247244&HistoricalAwards=false
• 关键词: Geometric; counts; surfaces

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.

表面是几何形状，到处都是二维的。这些几何对象在纯数学以及科学和工程的各个领域中发挥着重要作用。这个项目的重点是理解曲面的度量和几何属性，即长度和角度的概念。变形曲面的度量结构会产生所谓的“模空间”；模空间中的每个点表示下表面上的特定几何形状。PI将继续研究两类自然的曲面几何：(i)奇异平面度量，（ii）双曲（负弯曲）度量。对平面度量的理解的提高已经并将继续导致动力系统的新结果。从某种意义上说，双曲度规是这些曲面上最自然的度规。该项目还将研究曲面模空间上的各种结构，特别是测度，它提供了一个定义良好的“随机曲面”概念。PI将使用并进一步发展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。这个项目更广泛的影响包括产生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项目，在集中和协作的环境中研究问题。在平面度量设置下，PI利用新的多尺度紧化平动面地层，证明了遍历sl_2不变概率测度的强正则性，验证了仿射不变流形的自然启发。这一结果及其证明中的技术，已经被其他人应用于研究随机Teichmuller测地线和鞍连接对的计数。在另一项联合工作中，PI证明了在多尺度边界附近可以定义仿射不变流形的方程类型的限制，从而提供了Wright圆柱体变形定理的新证明，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，研究仿射不变流形的中心重要分类问题，以及关于k2的较少研究的k微分的问题。PI还开发了一个关于随机双曲曲面的研究项目，这是一个年轻而活跃的领域。与Sapir一起，PI证明了一个关于各种随机曲面的简单和非简单封闭测地线的相对数量的猜想，因为属趋于无穷。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已经解决并将继续调查硬币两面的问题。这项研究引出了关于一般曲面上典型测地线形状的进一步问题，以及关于每一个双曲曲面上的测地线的某些问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。