2012 Redbud Geometry/Topology Conference

这次会议将以李涛和其他专家关于Heegaard分裂和基本群等级的一些演讲为特色。Heegaard分裂是将三维流形分解为沿曲面沿着的两个简单部分，称为平行体。Heegaard分裂定义了3-流形的基本群的表示，其中生成元的数量等于曲面的亏格。因此，3-流形的秩（生成元的最小数目）小于或等于最小Heegaard曲面的亏格。在大多数情况下，秩和属是相等的，但也有少数例子发现秩严格小于属。李涛最近发现了秩严格小于亏格的双曲三维流形的第一个例子。这就提出了一个问题，这些例子有多普遍，以及等级和属之间的差异有多大。当我们从地面看地球表面时，它似乎是一个平面，永远向各个方向延伸。然而，从世界导航和从太空观看地球，我们知道它实际上是一个完全不同的形状-一个球体，或球的表面。同样，我们生活的宇宙似乎是一个平坦的三维空间，永远向各个方向延伸。不幸的是，我们没有办法检查它实际上是什么形状，但我们可以研究它可能是什么形状。这是三维拓扑学的重点。本次会议将汇集来自世界各地的专家，讨论在研究三维代数方法和几何方法之间的接口的最新发展spaces.http://comp.uark.edu/~matthewd/redbud/http://www.math.okstate.edu/~jjohnson/redbud/

This conference will feature a number of talks by Tao Li and by other experts on Heegaard splittings and ranks of fundamental groups. A Heegaard splitting is a decomposition of a 3-dimensional manifold into two simple pieces called handlebodies along a surface. A Heegaard splitting defines a presentation for the fundamental group of the 3-manifold in which the number of generators is equal to the genus of the surface. Thus the rank of the 3-manifold (the minimal number of generators) is less than or equal to the genus of the minimal Heegaard surface. In most cases, rank and genus are equal, but a small number of examples have been found in which the rank is strictly less than the genus. Tao Li recently discovered the first examples of hyperbolic 3-manifolds with rank strictly less than genus. This raises the question of how common these examples are and how big the difference between rank and genus can be.When we look at the surface of the earth from ground level, it appears to be a flat plane that goes on forever in all directions. However, from world navigation and viewing the earth from space, we know that it is, in fact, a completely different shape - a sphere, or the surface of a ball. Similarly, the universe that we live in appears to be a flat 3-dimensional space that goes on forever in all directions. Unfortunately we have no way of checking what the shape actually is, but we can study the possible shapes that it could be. This is the focus of 3-dimensional topology. This conference will bring together experts from around the world to discuss recent developments at the interface between algebraic methods and geometric methods for studying 3-dimensional spaces.http://comp.uark.edu/~matthewd/redbud/http://www.math.okstate.edu/~jjohnson/redbud/