Topics in low-dimensional topology

长期计划工作的各种项目有关的几何，代数和结构方面的低和高维双曲，真实的投影和复杂的双曲流形。 这些问题形成了一个广泛的频谱，包括一个有趣的全新领域的探索：双曲流形的真实的投影变形。 这些想法与复杂的双曲等距群内的群变形有联系，人们希望这将对复杂的双曲流形的研究产生影响，而这几乎一无所知。 此外，提议者将建立在他最近的工作方面的问题连接到表面群的存在，以及如何可以增加一个双曲流形的同源性，包括一个新的群论的角度对这些问题，并纳入新的进展算术的情况。 最后，我们希望在一个项目上工作，该项目考虑伪Anosov映射作用于穿孔环面，它将来自椭圆曲线的线程绘制在一起，伪Anosov动力系统和数论。一个空间被称为3流形，如果它是由小块，所有这些都是“类似”的普通3-Perelmann最近的开创性工作表明，通过证明这些流形是“几何的”，人们可以对它们有某种全面的理解。 在几何流形中，最重要的是所谓的双曲流形。 这门课在数学的许多领域都是普遍存在的，从低维拓扑学到动力系统，数论。事实上，这门课在物理学中也非常重要。 然而，即使与Perlemann的工作，双曲流形本身仍然是相当不了解，虽然他们的重要性，使他们成为磁铁的研究远远超过三十年。 这一建议本身对双曲流形的结构研究方面，以及它们如何可以弯曲成其他类型的空间。 提出者的结果限制了在宇宙的某些物理模型中可能出现的流形的种类，这种类型的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.