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-流形的通用覆盖的双Lipschitz分类），这涉及了大量研究人员十多年。他还将使用双Lipschitz几何和结理论的工具，在寻找更灵活的方法来分类问题的复杂表面奇点。 对于双曲流形，流形的几何、表示论和量子不变量之间存在假设的联系，PI将致力于与这些不变量相关的具体描述。PI还将继续他的研究3-流形的可拓性。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数学的力量。具体的工具，将使用的是研究的几何有界失真（“双Lipschitz几何”）和使用代数和数论不变量。

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.