Three-dimensional topology and some four-dimensional contexts

沙尔曼最近的研究集中在理论的经典结和3-流形，特别是对使用Heegaard分裂和相关概念（如桥梁positionings，隧道数）从经典结理论。 重点通常是在3-流形（经典方法）中包含的曲面的行为，但以更复杂的方式。 除了关于曲面相交的老式组合论证之外，还有来自图论的思想;除了对三维流形上的层次结构的经典研究之外，还有Gabai的缝合流形分解的概念，其中参数化曲面和Thurston范数的估计有助于控制和理解层次结构的拓扑结构;并且，添加到莫尔斯理论的经典工具，是瘦位置的极小极大原则，其中给定索引的句柄不是一次全部添加，而是尽可能缓慢地添加。 最近Scharlemann感兴趣的是如何这些和类似的新工具也可以用来解决问题的拓扑结构的4流形。 例如，他最近证明属三Schoenflies猜想开始努力证明充分Schoenflies猜想，除其他想法，4维应用薄的立场。 亏格3的证明（包括一般问题的思想及其与性质R的联系）也通过Heegaard并的自然使用将Heegaard理论整合到这个重要的四维问题中。 这些可能只是第一步的一个有用的应用程序的3流形的想法，这些有趣的拓扑问题，有时被称为（3 + 1）维，因为他们问如何3和4维流形是相互关联的。一个重点Scharlemann的兴趣多年来一直是拓扑结构的3流形。 解释一下：关于我们周围世界的最基本的观察之一，几乎从我们出生时就很明显，就是它是三维的。 因此，理解具有这种性质的物体是很有趣的：任何生活在其中的人都会把他们的世界看作是三维的。 这样的物体被称为“三维流形”，这项研究提案的广泛目标是增加我们对它们的理解。 特别（但不是唯一）的兴趣是我们对三维流形的新认识可以告诉我们关于四维流形的一些古老而重要的问题。 当我们将时间和空间纳入我们的思维时，这种四维流形也与我们的自然经验相联系。 在这个建议中特别强调的是坐在3维和4维流形理论之间的边缘的问题。 这两个维度都很有趣，部分原因是它们模拟了我们生活的宇宙。

Scharlemann's recent research has centered on the theory of classical knots and of 3-manifolds, in particular on the use of Heegaard splittings and of related notions (eg bridge positionings, tunnel number) from classical knot theory. The focus is typically on the behavior of surfaces contained in the 3-manifolds (a classical approach) but in a more sophisticated way. Added to old-fashioned combinatorial arguments on surface intersections are ideas from graph theory; added to the classic study of hierarchies on 3-manifolds is Gabai's notion of sutured manifold decomposition, in which parameterizing surfaces and estimates of the Thurston norm help control and understand the topology of the hierarchy; and, added to the classic tool of Morse theory, is the minimax principle of thin position, in which handles of a given index are added not all at once, but as slowly as possible. Recently Scharlemann has gotten interested in how these and similar new tools can also be used towards resolving questions in the topology of 4-manifolds. For example, his recent proof of the genus three Schoenflies Conjecture began with an effort to prove the full Schoenflies Conjecture with, among other ideas, a 4-dimensional application of thin position. The proof of the genus three case (which includes ideas on the general problem and its connection to Property R) also integrates Heegaard theory into this important 4-dimensional problem through the natural use of Heegaard unions. These may be only the first steps of a useful application of 3-manifold ideas to those intriguing topological questions which are sometimes called (3 + 1)-dimensional because they ask how 3- and 4-dimensional manifolds are interrelated. A focus of Scharlemann's interest for many years has been the topology of 3-manifolds. To explain: one of the most basic observations about the world around us, apparent almost from our birth, is that it is 3-dimensional. So it is of interest to understand objects with precisely this property: anyone living in one would see their world as 3-dimensional. Such objects are called ``3-manifolds", and the broad goal of this research proposal is to increase our understanding of them. Of particular (but not sole) interest is what our emerging understanding of 3-dimensional manifolds can tell us about some old and important questions concerning 4-dimensional manifolds. Such 4-manifolds also connect to our natural experience, when we incorporate time as well as space into our thinking. The particular emphasis in this proposal is on questions that sit on the edge between 3- and 4-dimensional manifold theory. Both of these dimensions are interesting in part because they model the universe in which we live.