Research in Geometry and Topology

9971404Bestvina这个项目的目标是理解三维庞加莱对偶群和对、映射类群的子群和自由群的自同构、复双曲曲面、对称空间中的多边形、线排列、Artin群和群上的维函数。该项目部分关注几何和代数中自然产生的物体的对称结构。古希腊人理解柏拉图立体的对称性，例如立方体或十二面体。柏拉图立体是正弯曲的，它们有无限的对称性。在光谱的另一端是负弯曲的物体，比如罗巴切夫斯基的双曲平面（在埃舍尔的画中经常出现），这些物体通常具有无限多的对称性。因此，这些对称的结构既有趣又难以理解。一个物体的对称构成一个“群”。衡量一个群体的“大小”的一种方法，即使这个群体是无限的，也是谈论它的“维度”。例如，上面提到的埃舍尔画中双曲平面的对称群的维数是2，反映了平面的性质。一般来说，尺寸的概念取决于背景编号系统（系数环）的选择。该项目将研究这种依赖性。该项目的另一部分是对多边形的研究。在欧几里得平面和空间中，这些看似简单的物体随着人们考虑更复杂的空间而变得越来越复杂。一个新现象是多边形的边不仅可以由它的长度决定，而且可以由它的方向决定。“在普通空间中，如果一个人固定了一条边的原点，他可以旋转另一端以获得任意方向。欧几里得三角形由它的三条边长决定。要构造这样一个三角形，就必须满足三个三角形不等式。“这些性质在更一般的空间中不再成立，这使得多边形的研究更加有趣和复杂。

9971404Bestvina The goal of this project is to understand 3-dimensional Poincare duality groups and pairs, subgroups of mapping class groupsand automorphisms of free groups, complex-hyperbolic surfaces,polygons in symmetric spaces, line arrangements, Artin groups, anddimension functions on groups. The project focuses in part on the structure of symmetries ofobjects naturally arising in geometry and algebra. The ancient Greeksunderstood symmetries of Platonic solids, e.g., a cube or adodecahedron. Platonic solids are positively curved, and they have afinite number of symmetries. At the opposite end of the spectrum arenegatively curved objects, such as the Lobachevsky hyperbolic plane(featured frequently in Escher paintings), and these typically haveinfinitely many symmetries. Consequently, the structure of thesesymmetries is both more interesting and harder to understand. Thesymmetries of an object form a ``group.'' One way to measure the``size'' of a group, even when the group is infinite, is to talk aboutits ``dimension.'' For example, the dimension of the group ofsymmetries of the hyperbolic plane, depicted in the Escher drawingmentioned above, is two, reflecting the nature of the plane. Ingeneral, the concept of dimension depends on the background choice ofnumbering system (coefficient ring). The project will study thisdependence. Yet another part of the project is the study of polygons.These seemingly simple objects in the Euclidean plane and space becomeincreasingly more complex as one considers more complicated spaces. Oneof the new phenomena is that an edge of a polygon may be determined notjust by its length but also by its ``direction.'' In the case ofordinary space, if one fixes the origin of an edge, one can rotate thesecond end to attain an arbitrary direction. A Euclidean triangle isdetermined by its three side-lengths. To construct such a triangle itis enough to satisfy three ``triangle inequalities.'' These propertiesno longer hold in more general spaces, which makes study of polygonsthere much more interesting and complicated.***