Groups in Geometry and Topology

[摘要]奖项：dms -0905802首席研究员：Michael Kapovich本提案是对Kapovich前几年在几何、拓扑和几何群理论领域研究的延续。计划研究的主题大多围绕着群体在各种空间上的几何作用和流形上的几何结构，以及建筑的几何。卡波维奇计划的研究涵盖了建筑几何及其在表示理论中的应用，特别是，卡波维奇打算继续他对对称空间和建筑中与代数群相关的多边形连杆的模空间的几何研究。其他研究课题包括代数群中的离散群（特别是算术格的相干问题）、实射影和复射影结构。群体自然地以数学和物理对象的对称形式出现，就像墙壁图案、矿物、雪花，最终是整个宇宙。本课题研究群的代数性质与群表现为对称的空间几何之间的关系。这种双向关系对群论（代数的一部分）和几何都是有益的。“建筑物”理论提供了这种关系的一个例子。比较在建筑物中导航的各种方式，可以用某些不平等来数学地描述。这种不平等反过来又提供了关于建筑对称群的各种问题的答案。

AbstractAward: DMS-0905802Principal Investigator: Michael Kapovich This proposal is a continuation of Kapovich's research ofprevious years in the areas of geometry, topology and geometricgroup theory. Most subjects of the planned research revolvearound geometry of group actions on various spaces and geometricstructures on manifolds, as well as geometry of buildings. Theresearch planned by Kapovich covers geometry of buildings withapplications to representation theory, in particular, Kapovichintends to continue his study of the geometry of the modulispaces of polygonal linkages in symmetric spaces and buildings inrelation to the algebraic groups. Other topics of researchinclude discrete groups in algebraic groups (in particular, thecoherence problem for arithmetic lattices), real-projective andcomplex-projective structures.Groups appear naturally as symmetries of mathematical andphysical objects, like wall-patterns, minerals, snowflakes and,ultimately, the entire universe. This project studies relationbetween algebraic properties of groups and geometry of spaces forwhich groups appear as symmetries. This two-way relation isbeneficial for both group theory (which is a part of algebra) andgeometry. An example of such relation is provided by the theoryof "buildings." Comparing various ways to navigate in buildingscan be mathematically described in terms of certain inequalities.Such inequalities in turn provide answers to various questionsabout symmetry groups of buildings.