Groups in Geometry and Topology

这笔拨款支持的大部分研究方向都围绕着各种空间上的群体作用的几何和流形上的几何结构，以及建筑物的几何。更详细地说，M.Kapovich的研究涉及：(1)建筑几何及其在表象理论中的应用。Kapovich打算继续研究对称空间和建筑物中多边形连杆的模空间相对于代数群的几何。这项研究的一部分是在欧几里德建筑上建立热带结构。(2)高维Kleian群：Kapovich将研究高维Klein群的有限性质。(3)复射影变种的基本群：Kapovich将研究奇点为正交的不可约复射影变种的基本群。(4)Kapovich将研究直角Artin群在圆的对数同胚群中的嵌入。(5)TeichMuller空间的半双曲性：Kapovich将研究TeichMuller空间的粗非正曲率性质。(6)Kapovich将研究具有平凡第二同伦群的闭四维流形M的基本群的分类，这项研究的目的是更好地理解几何与群之间的相互作用。组自然地显示为自然几何对象的对称性。这种对称性的简单例子来自于墙纸瓷砖(其中几何是欧几里得的)或瓷砖出现在埃舍尔的一些图片(双曲几何)。此外，群表现为物理性质的几何对象的对称性，从基本粒子到整个宇宙(被视为几何对象)。代数(群论)允许一个人编码基本的对称性，相反，几何允许一个人成功地处理纯代数问题。这种看似不同的数学领域的相互作用的一个例子是项目(3)，它的目的是将三维双曲空间的对称性群应用于复代数几何(后者处理具有复变量的多项式方程组解空间的几何)。

Most directions of the research supported by this grant revolve around geometry of group actions on various spaces and geometric structures on manifolds, as well as geometry of buildings. In more details, research conducted by M. Kapovich concerns: (1) Geometry of buildings with applications to representation theory. Kapovich intends to continue his study of the geometry of the moduli spaces of polygonal linkages in symmetric spaces and buildings in relation to the algebraic groups. Part of this study is establishing tropical structures on Euclidean buildings. (2) Kleinian groups in higher dimensions: Kapovich will study finiteness properties of higher-dimensional Kleinian groups. (3) Fundamental groups of complex-projective varieties: Kapovich will study fundamental groups of irreducible complex-projective varieties whose singularities are normal crossings. (4) Kapovich will study embeddings of Right-angled Artin Groups in the group of diffeomorphisms of the circle. (5) Semihyperbolicity of Teichmuller space: Kapovich will study coarse nonpositive curvature properties of Teichmuller space. (6) Kapovich will study classification of fundamental groups of closed 4-dimensional manifolds M with trivial 2nd homotopy group.The goal of this research is to understand better interaction between geometry and groups. Groups appear naturally as symmetries of natural geometric objects. Simple examples of such symmetries come from wall-paper tilings (where geometry is Euclidean) or tilings appearing is some of the Escher pictures (hyperbolic geometry). Furthermore, groups appear as symmetries of geometric objects of physical nature, from elementary particles to the entire universe (treated as a geometric object). Algebra (group theory) allows one to encode the underlying symmetries, and, conversely, geometry allows one to approach successfully purely algebraic problems. One of the examples of such interaction of seemingly different mathematical fields is project (3), which aims to apply groups of symmetries of 3-dimensional hyperbolic space to complex-algebraic geometry (the latter deals with geometry of solution spaces of systems of polynomial equations with complex variables).