Topics in low-dimensional topology

Long计划从事与低维和高维双曲、实射影和复杂双曲流形的几何、代数和结构方面有关的各种项目。这些问题形成了一个广泛的范围，包括对一个有趣的全新领域的探索：双曲流形的真实投影变形。这些想法与复杂双曲等距群内的群变形有关，这有望对几乎一无所知的复杂双曲流形的研究产生影响。此外，建议者将建立在他最近在与曲面群的存在性和如何增加双曲流形的同调性有关的问题方面的工作，包括对这些问题的新的群论观点，并在算术情况下纳入新的进展。最后，我们希望在一个项目上工作，该项目考虑作用于穿孔环面的伪阿诺索夫映射，它将来自椭圆曲线，伪阿诺索夫动力系统和数论的线程聚集在一起。如果一个空间是由小块组成的，那么它就被称为3流形，这些小块都“像”我们生活的普通三维空间。佩雷尔曼最近的开创性工作表明，人们可以通过证明流形是“几何的”来获得对这种流形的某种整体理解。在几何流形中，到目前为止最重要的一类是双曲流形。这门课在数学的许多领域都很普遍，从低维拓扑，到动力系统，数论。事实上，这门课在物理学中也是至关重要的。然而，即使有了Perlemann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了30多年来研究的磁石。这一建议指向双曲流形的结构研究方面，以及它们如何被弯曲成其他类型的空间。提出者的结果限制了流形的种类这些流形可以出现在宇宙的某些物理模型中这种弯曲在物理学中很重要。他还打算继续研究一个著名的古老猜想，即哪些二维物体可以存在于这些三维物体中。

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.