Groups in Geometry and Topology

AbstractAward：DMS-0905802首席研究员：Michael Kapovich这个提案是Kapovich前几年在几何、拓扑和几何群论领域研究的延续。计划研究的大多数主题围绕着各种空间上的群作用几何和流形上的几何结构，以及建筑物的几何。 该研究计划涵盖的Kapovich几何的建筑物与应用的代表性理论，特别是Kapovich打算继续他的研究几何modulispaces的多边形联系在对称空间和建筑物的关系代数群。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。群自然地表现为数学和物理对象的对称性，如墙壁图案、矿物、雪花，最终表现为整个宇宙。 本计画研究群的代数性质与群呈现对称性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）和几何都有好处。这种关系的一个例子是“建筑物”理论。“比较建筑物中的各种导航方式可以用某些不等式进行数学描述。这些不等式反过来又为关于建筑物对称群的各种问题提供了答案。

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.