3-manifolds and geometry

PI将与Behrstock合作完成3流形群的准等长分类（或等效地，3流形的泛覆盖的bi-Lipschitz分类），这已经涉及了大量的研究人员超过十年。他还将使用双lipschitz几何和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、表示理论和基于量子的不变量之间存在假定的联系，PI将致力于有关这些不变量的具体猜想。PI还将继续研究3流形的可通约性性质。这个建议解决了有关三维空间形式的问题，这些问题来自不同的数学领域。这些问题创造了低维拓扑与代数、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提供了数学的大部分力量，加强这些联系会增加这种力量。具体的工具，将使用几何的研究，直到有界畸变（“双利普希茨几何”）和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.