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; 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.

该项目在促进我们对数学的理解和建立国家科学和技术劳动力方面都有指示。该数学部分旨在提高我们对最有效地导航环境时表面形状的理解。当然，有效的概念取决于上下文，因此项目考虑了许多设置，希望在最佳形状中发现差异和相似之处，因为“最佳形状”的标准会改变。 在教育方面，这里的环境是，在未来十年中，国家将需要比我们预期的，该工程师比目前配置的管道要多得多。同时，来自资源较低的高中的学生，即使聪明，勤奋并对科学，技术，工程或数学的职业感兴趣，也以惊人的速度离开了这些STEM领域，因为他们很难从高中过渡到大学。由PI领导的计划在减少具有高潜力但不太理想的准备工作的STEM学生的流失方面取得了显着的成功：该赠款将有助于成长，维持，发展，发展和传播有关这种全面的整体方法来保留STEM中学生的信息。 该项目将通过谐波图研究在几个低等级谎言基团的Hitchin组分中表面组表示的渐近尸体。从表面到相关的对称空间的模棱两可的谐波图具有全体形态不变性，其几何拓扑可以预测代表的载体，直到衰减误差。 同时，错误估计值足够强，可以提出一种方法的统一性：对范围和地图的重新缩放为建筑物产生谐波图，而显然不同的建筑物可以通过相关的实际封闭场和估值来代数构造。其他项目包括通过模量空间技术在三个空间中找到一个新的基本最小表面，通过几何分析技术的新型统一度量度量，以及对表面上经典圆圈包装的改进。 PI将继续他对本科生，研究生和博士后学者的指导。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估的评估来支持的。