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

双曲空间之间的调和映射、将数域实现为不变迹域以及在双曲群中构造曲面子群

• 批准号: 1500951
• 负责人: Vladimir Markovic
• 金额: $ 42万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500951&HistoricalAwards=false
• 关键词: Harmonic; Maps; between; Hyperbolic; Spaces

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.

纯数学促进了思想的发展，这些思想后来被用于自然科学，如物理和生物学。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被证明对回答基本的物理问题很重要。该项目研究和开发了涉及粗双曲几何和各种几何流动的统计特性之间的相互作用的技术，这些技术正在成为证明来自一系列领域的结果的不可或缺的工具。尽管这种流动的几何学和动力学并不总是必要来陈述或证明这些突破，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。通过参与这个研究项目，下一代研究生将被介绍给这些概念。PI将学习关于双曲群和负曲线流形的几何问题。结合Cannon猜想，讨论了边界为2-球面的双曲群是否含有丰富的拟凸曲面子群的问题。在不同的方向上，我们将研究寻找具有给定拟对称边值的调和映象的Dirichlet型问题，特别是对Schoen猜想的研究。