Harmonic Maps between Hyperbolic Spaces, Realizing Number Fields as Invariant Trace Fields, and Constructing Surface Subgroups in Hyperbolic Groups

纯数学促进了后来在物理学和生物学等自然科学中使用的思想的发展。在物理学中，宇宙被描述为一个三维空间;研究三维流形的几何和拓扑可能对回答基本物理问题很重要。该项目研究和开发涉及粗糙双曲几何和各种几何流的统计特性之间相互作用的技术，这些技术正在成为证明一系列领域结果的不可或缺的工具。虽然这种流动的几何形状和动力学并不总是陈述或证明这些突破所必需的，但它们给出了思考这些突破的“正确”方式，并可能催化进一步的进展。通过参与该研究项目，下一代研究生将被引入这些概念。PI将研究双曲群和负弯曲流形的几何问题。与Cannon猜想有关，我们将讨论一个以2-球面为边界的双曲群是否包含大量的拟凸曲面子群的问题。在不同的方向，狄利克雷型问题找到调和映射与规定的拟对称边界值将研究，特别强调的Schoen猜想。

Pure mathematics fosters the development of ideas that are later utilized in natural sciences such as physics and biology. In physics, the universe is described as a 3-dimensional space; studying the geometry and topology of 3-manifolds may prove important in answering fundamental physical questions. This project investigates and develops techniques that involve an interplay between coarse hyperbolic geometry and statistical properties of various geometric flows, which are becoming indispensable tools in proving results from a range of fields. Although geometry and dynamics of such flows are not always necessary to state or prove these breakthroughs, they give the 'right' way of thinking about them and are likely to catalyze further progress. Through participation in this research project, the next generation of graduate students will be introduced to these concepts.The PI will study questions about geometry of hyperbolic groups and negatively curved manifolds. In connection with the Cannon Conjecture, the question of whether a hyperbolic group whose boundary is the 2-sphere contains an abundance of quasi-convex surface subgroups will be addressed. In a different direction, a Dirichlet type problem of finding harmonic mappings with prescribed quasi-symmetric boundary values will be studied, with particular emphasis on the Schoen Conjecture.