Groups in Geometry and Topology

这项资助支持的研究的大部分方向围绕几何群体行动的各种空间和几何结构的流形，以及几何建筑物。更详细地说，由M。Kapovich关注：（1）建筑物的几何与应用表示论。卡波维奇打算继续他的研究几何的模空间的多边形联系在对称空间和建筑物的关系代数群。这项研究的一部分是建立在欧几里得建筑热带结构。(2)Kleinian群在更高的维度：Kapovich将研究有限性的高维Kleinian群。(3)基本群体的复杂投射品种：卡波维奇将研究基本群体的不可约复杂投射品种的奇异性是正常的交叉。(4)卡波维奇将研究嵌入直角阿廷集团集团的迪#64256;eomorphisms的循环。(5)Teichmuller空间的半双曲性：Kapovich将研究Teichmuller空间的粗糙非正曲率性质。(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。这项研究的目标是更好地理解几何和群之间的相互作用。组自然地表现为自然几何对象的对称性。这种对称性的简单例子来自墙纸拼贴（其中几何是欧几里德）或拼贴出现在一些更复杂的图片（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）允许人们对基本的对称性进行编码，相反，几何允许人们成功地处理纯粹的代数问题。这种看似不同的数学领域之间的相互作用的一个例子是项目（3），其目的是将三维双曲空间的对称群应用于复代数几何（后者涉及复变量多项式方程组的解空间的几何）。

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).