Three-dimensional topology and some four-dimensional contexts

Scharlemann最近的研究集中在经典纽结和三维流形理论上，特别是Heegaard分裂和经典纽结理论中的相关概念(如桥位置、隧道数)的使用。重点通常是包含在三维流形中的曲面的行为(经典方法)，但以更复杂的方式。除了关于曲面交点的老式组合论证之外，还有来自图论的思想；除了对3-流形上的层次的经典研究之外，还有Gabai的缝合流形分解的概念，其中参数化曲面和瑟斯顿范数的估计有助于控制和理解层次的拓扑；以及，添加到Morse理论的经典工具中的是薄位置的极小极大原理，其中给定索引的句柄不是一次全部添加，而是尽可能慢地添加。最近，Scharlemann对如何使用这些和类似的新工具来解决4-流形拓扑中的问题产生了兴趣。例如，他最近对三个勋蝇猜想的证明，首先是试图用薄位置的四维应用来证明整个勋蝇猜想。亏格三情形的证明(包括关于一般问题及其与性质R的联系的思想)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3-流形思想在那些有趣的拓扑问题上有用应用的第一步，这些问题有时被称为(3+1)维，因为它们询问三维和4维流形是如何相互关联的。Scharlemann多年来一直感兴趣的一个焦点是3-流形的拓扑。解释：关于我们周围世界的最基本的观察之一，几乎从我们出生起就很明显，就是它是三维的。因此，准确地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认为他们的世界是三维的。这样的物体被称为‘3-流形’，这项研究计划的广泛目标是增加我们对它们的理解。特别(但不是唯一)感兴趣的是，我们对3维流形的理解可以告诉我们一些关于4维流形的古老而重要的问题。当我们将时间和空间融入我们的思维时，这样的4-流形也与我们的自然经验有关。在这个建议中，特别强调的是位于3维和4维流形理论之间的问题。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.