Mathematical Sciences: Seifert Fiber Spaces

Seifert流形是一种具有束状结构的拓扑空间；也就是说，它有一个参数空间，在参数空间中的每个点上都有另一个称为“纤维”的空间。纤维在参数空间中的点与点之间略有不同。这类空间自然地出现在许多不同的数学分支中，例如拓扑学、微分几何和代数几何，以及物理学(结晶学群)。调查者将研究这些空间的几何和拓扑以及它们的对称性。所要研究的问题从关于这些空间的几何结构的一般问题到与对称和非对称Seifert流形的存在有关的具体问题。这将极大地提高我们对这类重要空间及其内在和外在几何、结构和性质的认识。本研究将使用的技术和方法包括代数和几何拓扑学、微分几何、李群理论和群上同调等方法。

A Seifert manifold is a topological space with a bundle-like structure; that is, it has a parameter space and over every point in the parameter space, there sits another space called "a fiber." The fibers vary slightly from point to point in the parameter space. Spaces of this class occur naturally in many different branches of mathematics, for example, topology, differential geometry and algebraic geometry, as well as in physics (crystallographic groups). The investigator will study the geometry and topology of these spaces as well as their symmetries. The problems to be studied vary from general questions concerning the geometric structures of these spaces to specific problems connected with the existence of symmetric and asymmetric Seifert manifolds. This will greatly enhance our knowledge of this important class of spaces, their intrinsic and extrinsic geometry, structure and properties. The broad spectrum of techniques and methods which will be employed in this research include methods of algebraic and geometric topology, differential geometry, Lie group theory and group cohomology.