CAREER: Finer Coarse Geometry

本提案的研究主题以几何群论和几何拓扑为中心，但与其他几个领域有联系，包括数论、组合学、凸几何、动力学和概率论。几何群论最常关注的是准等距不变量：对距离的有界（加性和乘性）畸变不敏感的测量。这允许我们在群的研究和它们在几何上作用的空间之间，以及有限生成群的不同Cayley图之间传递。这里描述的研究项目的一个统一主题是通过使用更灵敏的测量来追求“更精细”的渐近几何方法。特别地，PI提出研究在准等距下不不变的大规模几何统计量。这种观点有助于研究群中的随机性和渐近密度，以及空间中“典型”测地线的性质。这里的想法已经让PI和她的合作者在自由阿贝尔群和海森堡群中得到了新的密度结果；用新的不变量进一步完善直角Artin群的分类程序；以及特赫穆勒几何研究的进展。统计几何广泛应用于纯数学之外的一系列领域，从计算机科学到医学——只要是建立几何模型和需要长期概率预测的领域。除了研究计划之外，本提案还描述了一系列教育项目：主要是研究实验室组织基础设施的发展，吸引来自各种机构的教师和学生。每个实验室将围绕一个连贯的数学核心研究一组问题，并提供多层次教学、学习、探索和合作的机会。将有与这些实验室协调的会议和期刊发行，以说明性与研究价值相结合的原则为指导，强调合作。

The research themes in this proposal are centered in geometric group theory and geometric topology, but have connections to several other fields, including number theory, combinatorics, convex geometry, dynamics, and probability. Geometric group theory is most often focused on quasi-isometry invariants: measurements that are insensitive to bounded (additive and multiplicative) distortion of distances. This allows us to pass between the study of groups and the spaces they act on geometrically, and between different Cayley graphs for finitely generated groups. A unifying theme for the research projects described here is the pursuit of "finer" approaches to asymptotic geometry through the use of more sensitive measurements. In particular, the PI proposes to study large-scale geometric statistics that are not invariant under quasi-isometry. This point of view facilitates the study of randomness and asymptotic density in groups, and of properties of "typical" geodesics in spaces.The ideas in play here have already led the PI and her collaborators to new density results in free abelian groups and the Heisenberg group; new invariants used to further the classification program for right-angled Artin groups; and progress in the study of Teichmueller geometry. Statistical geometry is broadly applicable in a range of settings outside pure mathematics, from computer science to medicine---wherever geometric models are made and long-term probabilistic prediction is required. In addition to the research program, this proposal describes a collection of educational projects: principally, the development of an organizational infrastructure for Research Labs engaging faculty and students from a wide variety of institutions. Each lab will work on a cluster of problems around a coherent mathematical nucleus, with opportunities for mutli-level teaching, learning, exploration, and collaboration. There will be conferences and journal issues coordinated with these Labs, guided by the principles of combining expository soundness with research value and emphasizing collaboration.