3-manifolds and geometry

PI将与Behrstock合作完成3-流形群的准等距分类(或等价地，3-流形的泛覆盖的双Lipschitz分类)，这涉及了大量研究人员十多年。他还将使用双Lipschitz几何和纽结理论工具来寻找更灵活的方法来解决复杂表面奇点的分类问题。对于双曲流形，流形的几何不变量、表示论不变量和基于量子的不变量之间存在假定的联系，PI将致力于与这些不变量相关的特定猜想。PI还将继续他对三维流形的可公度性的研究。这项建议解决了关于三维空间形式的问题，这些问题来自不同的数学领域。这些问题产生了低维拓扑学与代数、代数几何、数论和理论物理的相互作用。不同领域之间的联系在很大程度上提供了数学的力量，而加强这些联系会增加这种力量。将使用的具体工具是研究直到有界扭曲的几何学(“双Lipschitz几何学”)以及代数和数论不变量的使用。

The PI will work with Behrstock to complete the quasi-isometric classification of 3-manifold groups (or equivalently, the bi-Lipschitz classification of universal covers of 3-manifolds), which has involved a large number of researchers for over a decade. He will also use bi-Lipschitz geometry and knot-theoretic tools in the search for more flexible approaches to classification questions for complex surface singularities. For hyperbolic manifolds there are postulated connections between geometric, representation-theoretic, and quantum based invariants of manifolds, and the PI will work towards specific conjectures relating these invariants. The PI will also continue his investigation of commensurability properties of 3-manifolds.This proposal addresses questions concerning 3-dimensional space forms that arise from disparate areas of mathematics. These questions create interactions of low dimensional topology with algebra, algebraic geometry, number theory, and theoretical physics. Links between disparate areas provide much of the power of mathematics, and strengthening these links increases the power. Specific tools that will be used are the study of the geometry up to bounded distortion ("bi-Lipschitz geometry") and the use of algebraic and number-theoretic invariants.