Geometric Variational Problems in Classical and Higher Rank Teichmuller theory

这个项目在增进我们对数学的理解和建设国家科学技术队伍方面都有方向。数学部分的目的是促进我们对曲面在环境中最有效地导航时呈现的形状的理解。当然，效率的概念取决于背景，因此该项目考虑了许多设置，期望在最佳形状的标准改变时找到最佳形状的差异和相似之处。在教育方面，目前的情况是，未来十年，国家需要的工程师将比我们预期的管道--目前的配置--多出100万人。与此同时，资源不是很充足的高中的学生，即使聪明勤奋，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STEM领域，因为他们在从高中过渡到大学方面遇到了困难。由PI牵头的一项计划在减少STEM学生的自然减员方面取得了显著成功：这笔赠款将帮助增长、维持、发展和传播关于留住STEM学生的全面全面方法的信息。该项目将通过调和映射来研究几个低阶李群的Hitchin分量中表面群表示的渐近完整。从曲面到伴随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，直到衰减误差。与此同时，误差估计足够强大，足以表明方法的统一性：范围和地图的重新缩放产生到建筑的调和映射，而明显不同的建筑可以通过相关的实闭域和估值以代数方式构建。其他项目包括通过模空间技术在三维空间中寻找新的基本极小曲面，通过几何分析技术找到一种新型的均匀度规，以及对曲面上的一个经典的圆填充结果进行精化。PI将继续指导本科生、研究生和博士后奖学金。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.