Heegaard Splittings and the Combinatorics of Three-Manifolds

PI试图理解三流形的算法结构。准确地说：PI将致力于Heegaard亏格问题，并将攻击计算Heegaard图的Hempel距离的问题。 这两个问题都很难，因为它们与庞加莱猜想和三球识别问题有关。 有几个现有的工具，其中春天的想法：一方面哈肯和鲁宾斯坦的组合理论的正常和几乎正常的surfaceswhile另一方面有想法从理论的Kleiniangroups，涉及曲线复杂。给味道的算法问题涉及考虑以下“玩具”版本：你的花园软管，躺在草坪上，是非常纠结。 在把它收起来之前，你必须把它解开。 是否有一个机械程序（编程到您的机器人管家），将解开软管无论其开始位置？ 手术有多快？ 如果邻居的孩子把水管的两端拧在一起，你的管家至少能把水管拉直成一个圆圈吗？如果水管短一点，管家能更快地完成吗？ 快多少？

The PI seeks to understand the algorithmic structure of three-manifolds.To be precise: the PI will work on the Heegaard genus problem and willattack the question of computing the Hempel distance of a Heegaarddiagram. Both questions are difficult due to their connections to thePoincare Conjecture and the three-sphere recognition problem. There areseveral existing tools which spring to mind: on the one hand Haken andRubinstein's combinatorial theory of normal and almost normal surfaceswhile on the other hand there are ideas from the theory of Kleiniangroups, involving the curve complex.To give the flavor of the algorithmic problems involved consider thefollowing "toy" version: your garden hose, lying on the lawn, is verytangled. You must untangle it before putting it away. Is there amechanical procedure (to program into your robot butler) which willuntangle the hose regardless of its starting position? How fast is theprocedure? If the neighbor's child screws the ends of the hose togethercan your butler at least manage to straighten the hose into a circle?Can the butler finish faster if the hose is shorter? How much faster?