Mathematical Sciences: Essential Laminations in 3-Manifolds

9400651 Brittenham本质分层概括了对3-流形拓扑重要的两个对象：不可压缩曲面和紧致叶理。这两个更经典的对象在过去都被证明是非常有用的(今天仍然是如此)，而本质薄片最近在解决三维流形理论中的许多基本问题方面表现出了类似的力量。这位研究者打算继续他关于本质叠层的工作，考虑到三个主要目标：(1)证明它们在许多情况下的行为很像它们更容易处理的近亲--不可压缩曲面；(2)证明包含本质叠层的同伦等价的3-流形是同胚的；以及(3)利用叠层的Haken范式的概念，在各种各样的3-流形中构造本质叠层。换句话说，研究者打算证明本质层(1)表现良好，(2)告诉我们关于3-流形的有趣的事情，(3)可以在‘大多数’3-流形中找到。令人惊讶的是，尽管我们生活在一个三维世界中，因此对事物在三维空间中如何工作有大量的自然直觉，但我们对三维流形的理解还远远不完整，三维流形是以我们的三维空间为模型的物体。它的许多基本问题，如著名的庞加莱猜想，尽管在更高维度的类似问题已经得到解决，但仍然没有得到解决。许多新技术已经被开发出来，而且还在继续开发，试图解开这个熟悉的、尽管神秘的维度。研究人员打算继续开发一种研究三维流形的新技术，该技术使用(很好理解的)二维对象来研究这些(不太理解的)三维对象。我们的想法是，把一个三维物体想象成一个由堆叠在一起的二维“薄片”组成的集合，我们有时可以使用我们的(更好的！)对维度2的直觉和理解，将解决3维问题的方法“缝合在一起”。***

9400651 Brittenham Essential laminations generalize two objects important to 3-manifold topology: the incompressible surface and the taut foliation. Both of these more `classical' objects have proved themselves to be very useful in the past (and continue to be so, today), and essential laminations have recently shown similar power in attacking many of the fundamental problems in the theory of 3-manifolds. The investigator intends to continue his work on essential laminations, with three main goals in mind: (1) to show that they behave in many situations much like their far more manageable cousin, the incompressible surface, (2) to show that homotopy equivalent 3-manifolds, which contain essential laminations, are homeomorphic, and (3) to construct essential laminations in a wide variety of 3-manifolds, using the notion of Haken normal form for laminations. In other words, the investigator intends to show that essential laminations (1) behave nicely, (2) tell us interesting things about 3-manifolds, and (3) can be found in `most' 3-manifolds. It is somehow surprising that, in spite of the fact that we live in a 3-dimensional world, and therefore have a great deal of natural intuition about how things `work' in 3 dimensions, our understanding of 3-dimensional manifolds, objects modelled on our 3-dimensional space, is far from complete. Many of its basic problems, such as the celebrated Poincare conjecture, remain unsolved, even though the analogous problems in higher dimensions have been settled. Many new techniques have been developed, and continue to be developed, to try to unravel this familiar, though mysterious, dimension. The investigator intends to continue to delevop a new technique for studying 3-dimensional manifolds which uses (well-understood) 2-dimensional objects to study these (not so well-understood) 3-dimensional ones. The idea is that by thinking of a 3-dimensional object as being made up of a collection of 2-dimensionsal `sheets' , stacked together, we can sometimes use our (even better!) intuition and understanding about dimension 2 to `stitch together' a solution to a 3-dimensional problem. ***