3-Manifold Geometry and Topology

在这笔拨款下，PI将研究与3流形曲面相关的问题。虚Haken猜想问的是是否有一个有限片的非球面封闭3流形的盖上有一个注入到基群上的封闭曲面。PI建议利用Kahn-Markovic和Wise最近的结果来研究这个猜想。这将产生相关猜想的分支，例如虚拟纤维猜想，并将极大地阐明3流形及其不变量的结构，例如扭曲亚历山大多项式。PI还提出研究Thurston范数，以及通过缝合流形层次在最小同调类中实现Thurston范数的各种曲面之间的关系。他希望利用这些关系来研究关于瑟斯顿范数及其与3流形其他不变量的关系的各种问题。三维空间的数学研究可以追溯到庞加莱的工作。由于经典物理学将我们的宇宙描述为一个三维空间，因此三维空间的分类是一项重要的数学努力，因为它可能对我们宇宙的整体结构产生影响。这个项目将探讨三维空间分类的各个方面。主要和最雄心勃勃的项目将研究三维空间内的二维物体，以及如何在相关的“覆盖空间”中解决这些问题。研究这些二维物体对有限三维空间的整体结构具有重要意义。该项目的技术将与数学的其他领域有关，如群论、几何和四维空间的研究。

Under this grant, the PI will work on problems related to surfaces in 3-manifolds. The virtual Haken conjecture asks whether any aspherical closed 3-manifold has a finite-sheeted cover which contains a closed surface which injects on the fundamental group. The PI proposes to work on this conjecture making use of recent results of Kahn-Markovic and Wise. This would have ramifications for related conjectures, such as the virtual fibering conjecture, and would greatly elucidate the structure of 3-manifolds and their invariants, such as twisted Alexander polynomials. The PI also proposes to study the Thurston norm, and the relation between various surfaces realizing the Thurston norm in a minimal homology class via sutured manifold hierarchies. He hopes to use these relations to study various questions about the Thurston norm and its relation to other invariants of the 3-manifold.The mathematical study of 3-dimensional spaces goes back to the work of Poincare. Because classical physics describes our universe to be a 3-dimensional space, the classification of 3-dimensional spaces is an important mathematical endeavor, since it may have ramifications for the global structure of our universe. This project will pursue various aspects of the classification of 3-dimensional spaces. The principal and most ambitious project will study 2-dimensional objects inside of 3-dimensional spaces, and how one may resolve these in related "covering spaces". Studying these 2-dimensional objects has implications for the global structure of finite 3-dimensional spaces. The techniques of the project will have relations to other areas of mathematics such as group theory, geometry, and the study of 4-dimensional spaces.