Hyperbolic manifolds from a contact geometry perspective

在Thurston的工作之后，我们知道3流形的世界有不同的风格，这取决于我们所研究的流形的几何形状。从另一个角度来看，我们知道每个3流形都承认一个接触结构。在接触结构领域有一个经典的二分法：它们可以是紧的，也可以是过度扭曲的。从数学的角度来看，前者是最有趣的。近20年来，三维流形上紧密接触结构的分类一直是数学研究的中心问题。第一个被理解的例子是3球、固体环面和透镜空间。现在有很好的技术来处理塞弗特流形，但双曲流形上的紧密接触结构的世界仍然很大程度上未被探索。在这个博士项目中，学生将开始处理双曲流形上紧密接触结构的分类问题。

After Thurston's work we know that the world of 3-manifolds comes in different flavors, depending on the geometry of the manifolds we are looking at. From a different perspective, we know that every 3-manifold admits a contact structure. There is a classic dichotomy in the realm of contact structures: they can be tight or overtwisted. The former ones are the most interesting from a mathematical perspective. The classification of tight contact structures on 3-manifolds has been at the center of the mathematical research for the last 20 years. The first well understood examples were the 3-sphere, the solid torus and lens spaces. There are now well established techniques to deal with Seifert manifolds, but the world of tight contact structure on hyperbolic manifolds is still largely unexplored. In this PhD project the student will start tackling the problem of classification of tight contact structures on hyperbolic manifolds.