3-Manifolds and Geometry

该项目的一个目标是完成三维流形群的准等距分类，在过去的五年中，PI和Behrstock在这方面取得了非常重大的进展。另一种是用新方法研究复杂曲面的奇异性。特别是，通过PI、Birbrair和Pichon的工作，bilipschitz几何在揭示精细不变量方面的重要作用已经变得清晰起来。PI将把他们最近对复杂表面芽的内度量的bilipschitz分类推广到外度量，并将其应用于诸如Lipschitz和Zariski等值性之间的关系以及交同调和基于解析的上同调理论之间的猜想对偶等长期存在的问题。还将开始将该方法扩展到o-极小结构中的高维和集合。国际和平研究所还将继续研究与卡森不变量猜想有关的问题，卡森不变量猜想是迈克尔·阿提亚爵士最初提出的一个较宽松问题的具体表述，假设与拓扑量子场论有关的拓扑不变量与代数几何中的解析不变量之间存在联系，这种联系源于物理学。在光谱的另一端，对于双曲流形，流形的几何、算术、表示理论和基于量子的不变量之间也存在假设的联系，PI和他的学生已经在这些不变量上取得了进展，PI将继续在这些不变量上工作。准等距几何和bilipschitz几何允许有限的变形量，它提供了一个框架，使原本依赖于选择的几何变得独特。这个框架在大范围内是有用的，例如，对于紧致流形的泛覆盖的几何研究，其中它将几何/拓扑学与几何群论联系在一起，在小范围内，它是研究代数集的局部几何的自然框架。在这两种情况下，三维流形都扮演着重要的角色。此外，通过流形的不变量，人们与代数、数论和理论物理之间存在着低维拓扑的相互作用。使用这些方法，该项目解决了跨越不同数学领域的长期开放的理论问题。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。

One aim of the project is to complete the quasi-isometric classification of 3-manifold groups, on which the PI and Behrstock made very significant progress over the last five years. Another is to use new methods to study singularities of complex surfaces. In particular, the important role of bilipschitz geometry in revealing fine invariants has become clear through work of the PI, Birbrair, and Pichon. The PI will extend their recent bilipschitz classification of inner metric of complex surface germs to the outer metric, and apply this to such long-standing problems as the relationship between Lipschitz and Zariski equisingularity and the conjectured dualities between intersection homology and analytically based cohomology theories. A start will be also be made on extending the approach also to higher dimensions and sets in o-minimal structures. The PI will also continue studies related to the Casson invariant conjecture, a concrete formulation of a looser question originally asked by Sir Michael Atiyah, postulating links, motivated out of physics, between topological invariants related to topological quantum field theories and analytical invariants in algebraic geometry. At the other end of the spectrum, for hyperbolic manifolds there are also postulated connections between geometric, arithmetic, representation-theoretic and quantum based invariants of manifolds, which the PI and his students have made progress on and on which the PI will continue to work.Quasi-isometric and bilipschitz geometry, which allows a limited amount of deformation, provides a framework in which geometries which would otherwise depend on choices become unique. This framework is useful both in the large, e.g., for the geometric study of universal covers of compact manifolds, where it connects geometry/topology with geometric group theory, and in the small, where it is the natural framework for the study of the local geometry of algebraic sets. In both situations three-dimensional manifolds play an important role. Moreover, through invariants of manifolds one has interactions of low dimensional topology with algebra, number theory, and theoretical physics. Using these approaches the project addresses long-term open theoretical questions that span different areas of mathematics. Links between disparate areas provide much of the power of mathematics, and strengthening these links increases the power.