Geometric Variational Problems in Classical and Higher Rank Teichmuller theory

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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.