Geometric Variational Problems in Classical and Higher Rank Teichmuller theory

这个项目在提高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。数学部分旨在提高我们对表面形状的理解，当它们最有效地导航它们的环境时。当然，效率的概念取决于环境，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，这个国家在未来十年需要的工程师数量将超过我们目前预期的100万。与此同时，来自资源较差的高中的学生，即使他们聪明、勤奋，对科学、技术、工程或数学领域的职业感兴趣，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。由PI领导的一个项目在减少高潜力但未达到最佳准备的STEM学生的流失方面取得了显着成功：该赠款将帮助发展，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。本课题将通过调和映射，研究若干低秩李群的Hitchin分量中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，直至衰减误差。同时，误差估计足够强，可以提出统一的方法：重新缩放范围和地图产生建筑物的调和地图，而明显不同的建筑物可以通过相关的实封闭场和估值进行代数构建。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最小曲面，通过几何解析技术找到一种新的均匀度量，以及在曲面上改进经典的圆填充结果。PI将继续指导本科生、研究生和博士后学者。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。

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.