Geometry and Topology of Hyperbolic 3-Manifolds

双曲三维流形的研究在过去的五年中取得了巨大的进展。 现有的新工具有很大的潜力来解决以前无法解决的问题。 这些新的工具包括佩雷尔曼的里奇流技术，目前相当复杂的理论离散子群的PSL（2，C），并通过算术格与数论的相互作用。 Peter Storm将致力于将这些新工具应用于3-流形的研究，以及更一般的负弯曲空间，如Gromov双曲空间或高维负弯曲流形。三维流形是一个数学对象，它模拟了我们所占据的三维世界。 因此，关于三维空间性质的问题就变成了关于三维流形的数学问题。 由于这个和其他原因，三维流形被许多数学家仔细研究。 他们占据了一个特殊的利基之间的强烈直观的科目，如几何在平面上，和更抽象的科目，如研究数学对象与4个以上的维度，其中有意义的精神形象是很难产生。 Peter Storm的研究试图从直观和更抽象的观点中获益，目的是更好地理解三维物体及其数学表兄弟的性质。

The study of hyperbolic 3-manifolds has made tremendous progress in the past five years. The new tools available have great potential for attacking previously inapproachable questions. These new tools include Perelman's Ricci flow techniques, the presently quite sophisticated theory of discrete subgroups of PSL(2,C), and interactions with number theory via arithmetic lattices. Peter Storm will work to apply these new tools to the study of 3-manifolds, and more general negatively curved spaces such as Gromov hyperbolic spaces, or higher dimensional negatively curved manifolds. A 3-manifold is a mathematical object which models the three dimensional world we occupy. Thus questions about the nature of 3-dimensional space become mathematical questions about 3-manifolds. For this and other reasons, 3-manifolds are carefully studied by many mathematicians. They occupy a special niche between strongly intuitive subjects, such as geometry in the plane, and more abstract subjects, such as the study of mathematical objects with more than 4 dimensions, where meaningful mental images are difficult to produce. The research of Peter Storm tries to benefit from both the intuitive and the more abstract points of view, with the goal of gaining a better understanding the nature of three dimensional objects, and their mathematical cousins.