Topics in low-dimensional topology

Long计划致力于各种项目，涉及低维和高维双曲、实射影和复双曲流形的几何、代数和结构方面。这些问题构成了一个广泛的领域，包括对一个有趣的全新领域的探索：双曲流形的真实射影变形。这些概念与复双曲等距群内的群变形有关，人们希望这将对复双曲流形的研究产生影响，而复双曲流形的研究几乎一无所知。此外，提出者将以他最近关于与曲面群的存在有关的问题以及如何增加双曲流形的同调的方面的工作为基础，包括关于这些问题的新的群论观点，并在算术情况下结合新的进展。最后，我们希望开展一个项目，该项目考虑作用于穿孔环面的伪Anosov映射，它将来自椭圆曲线、伪Anosov动力系统和数论的线索结合在一起。如果一个空间是由小块组成的，那么它被称为3-流形，所有这些小块都与我们生活的普通3维空间一样。佩雷尔曼最近的开创性工作表明，人们可以通过证明这些流形是“几何的”来获得对这种流形的某种全局理解。在几何流形中，迄今为止最重要的是一类称为双曲流形的流形。从低维拓扑学到动力系统、数论，这个类在数学的许多领域都是普遍存在的。事实上，这个类在物理学中也是至关重要的。然而，即使有了佩尔曼的工作，人们对双曲流形本身的了解仍然相当少，尽管它们的重要性使它们成为研究的磁铁长达30多年。这一建议指向双曲流形的结构研究的各个方面，以及它们如何弯曲到其他类型的空间中。提出者的结果限制了在某些宇宙物理模型中可能出现的流形的种类，而这种类型的弯曲在物理学中是重要的。他还打算继续研究一个著名的古老猜想，关于哪些类型的二维物体可以在这些三维物体中生存。

Long plans to work on a variety of projects concerned with the geometric, algebraic and structural aspects of low and higher dimensional hyperbolic, real projective and complex hyperbolic manifolds. The problems form a broad spectrum including an exploration of an intriguing totally new area: the real projective deformations of hyperbolic manifolds. These ideas have connexions with group deformations inside complex hyperbolic isometry groups which it is hoped will have implications for the study of complex hyperbolic manifolds, about which almost nothing is known. Further, the proposer will build upon his recent work on aspects of the questions connected to the existence of surface groups and how one can increase the homology of a hyperbolic manifold, including a new group theoretic perspective on these issues and incorporating new progress in the arithmetic case. Finally, we hope to work on a project which considers pseudo-Anosov maps acting on a punctured torus, which draws together threads coming from elliptic curves, pseudo-Anosov dynamical systems and number theory.A space is called a 3-manifold if it is made of small chunks all of which are ``like'' the ordinary 3-dimensional space that we live in.The recent seminal work of Perelmann has shown that one can get a certain sort of global understanding of such manifolds by proving that they are ``geometric''. Amongst geometrical manifolds, by far the most important are the class called hyperbolic manifolds. This class is ubiquitous in many areas of mathematics, ranging from low-dimensional topology, to dynamical systems, number theory.Indeed, this class is also crucially important in physics. However, even with Perlemann's work, hyperbolic manifolds themselves are still fairly poorly understood, although their importance have made them a magnet for research for well over thirty years. This proposal directs itself towards aspects of the structural study of hyperbolic manifolds and how they can be bent into other types of spaces. The proposer has results which restrict the sorts of manifolds which can arise in certain physical models of the universe and this type of bending is imprtant in physics. He also intends to continue work on a famous old conjecture about which sorts of two-dimensional objects can live inside these 3-dimensional objects.