Geometric Variational Problems in Classical and Higher Rank Teichmuller theory

经典和高阶Teichmuller理论中的几何变分问题

• 批准号: 2005551
• 负责人: Michael Wolf
• 金额: $ 54.07万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005551&HistoricalAwards=false
• 关键词: Geometric; Variational; Problems; Classical; Higher

This project has directions both in term of advancing our understanding of mathematics and in building the nation's scientific and technical workforce. The mathematical part aims to advance our understanding of the shapes that surfaces present when they are most efficiently navigating their environment. Of course, the notion of efficient depends on the context, so the project considers a number of settings, expecting to find both differences and similarities in the optimal shapes as the criteria for "best shape" are changed. In terms of education, the setting is that nation will need about a million more engineers in the coming decade than we expect the pipeline, as it is currently configured, to produce. At the same time, students from less well-resourced high schools, even if smart and hard-working and interested in a career in science, technology, engineering or mathematics, leave those STEM fields at an alarming rate, as they have trouble transitioning from high school to college. A program led by the PI has achieved notable success in cutting the attrition from STEM students of high potential but less-than-optimal preparation: the grant will help grow, sustain, develop and disseminate information about this comprehensive holistic approach to retention of students in STEM. The project will investigate, via harmonic maps, the asymptotic holonomy of surface group representations in the Hitchin component of several low rank Lie groups. The equivariant harmonic maps from surfaces to the associated symmetric spaces have holomorphic invariants, the geometric topology of which can predict the holonomy of the representation, up to a decaying error. At the same time, the error estimates are strong enough to suggest a unity of approaches: a rescaling of the range and the maps produces a harmonic map to a building, while an apparently different building may be constructed algebraically via an associated real closed field and a valuation. Other projects include finding a new basic minimal surface in three-space through moduli space techniques, a new type of uniformized metric through geometric analytic techniques, and a refinement of a classical circle-packing result on surfaces. The PI will continue his mentorship of undergraduates, graduate students, and postdoctoral scholars.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.

这个项目在促进我们对数学的理解和建设国家的科学和技术劳动力方面都有方向。数学部分旨在促进我们对表面在最有效地导航其环境时呈现的形状的理解。当然，有效的概念取决于上下文，因此该项目考虑了许多设置，期望在“最佳形状”的标准发生变化时找到最佳形状的差异和相似之处。 在教育方面，我们的背景是，在未来十年里，国家将需要比我们预期的管道多出大约100万名工程师。与此同时，来自资源较少的高中的学生，即使聪明勤奋，对科学，技术，工程或数学感兴趣，也会以惊人的速度离开这些STEM领域，因为他们很难从高中过渡到大学。PI领导的一项计划在减少高潜力但准备不足的STEM学生的流失方面取得了显着的成功：该补助金将有助于增长，维持，发展和传播有关这种全面的整体方法的信息，以保留STEM学生。 该项目将调查，通过调和映射，在几个低秩李群的希钦分量的表面群表示的渐近完整性。从曲面到相关对称空间的等变调和映射具有全纯不变量，其几何拓扑可以预测表示的完整性，直到衰减误差。 与此同时，误差估计是强大的，足以表明一个统一的方法：范围和地图的重新缩放产生一个调和映射到一个建筑物，而一个明显不同的建筑物可以通过一个相关的真实的闭域和一个赋值代数构造。其他项目包括通过模空间技术在三维空间中找到一个新的基本极小曲面，通过几何分析技术找到一种新型的均匀化度量，以及对曲面上经典圆包装结果的改进。PI将继续指导本科生、研究生和博士后学者。该奖项反映了NSF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。

项目成果

HIGGS BUNDLES, HARMONIC MAPS, AND PLEATED SURFACES

希格斯束、调和图和褶皱表面

• 发表时间: 2024
• 期刊: JP Journal of Geometry and Topology
• 作者: Ott, Andreas;Swoboda, Jan;Wentworth, Richard;Wolf, Michael
• 通讯作者: Wolf, Michael

PLATEAU PROBLEMS FOR MAXIMAL SURFACES IN PSEUDO-HYPERBOLIC SPACE

伪双曲空间中最大曲面的平台问题

• 发表时间: 2024
• 期刊: Annales Scientifiques de lEcole Normale Supérieure
• 作者: Labourie, Francois;Toulisse, Jeremy;Wolf, Michael
• 通讯作者: Wolf, Michael

PLANAR MINIMAL SURFACES WITH POLYNOMIAL GROWTH IN THE Sp(4,R)-SYMMETRIC SPACE

Sp(4,R)对称空间中多项式增长的平面极小曲面

• 发表时间: 2025
• 期刊: American journal of mathematics
• 影响因子: 1.7
• 作者: Tamburelli, Andrea;Wolf, Michael
• 通讯作者: Wolf, Michael