Categorification and 3-Manifold Invariants in Algebraic Geometry

代数几何中的分类和 3 流形不变量

• 批准号: 1101439
• 负责人: Sabin Cautis
• 金额: $ 15万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101439&HistoricalAwards=false
• 关键词: Categorification; Manifold; Invariants; Algebraic; Geometry

The proposal is to construct 3-manifold invariants using tools and techniques from algebraic geometry and geometric representation theory. Given a Lie algebra one can define a numerical invariant of 3-manifolds called the Witten-Reshetikhin-Turaev invariant. It is an open problem to lift these to homological invariants of 3-manifolds. This is analogous to the way (singular) homology of topological spaces lifts the Euler characteristic. In earlier work of the PI jointly with Joel Kamnitzer they were able to do this for complements of links inside the 3-sphere when the Lie algebra is sl(n) and the link labeled by fundamental representations. To do this they used certain (derived) categories of coherent sheaves on flag-like varieties and the hope is to generalize this approach to other 3-manifolds. Their constructions were algebro-geometric but used many techniques from representation theory. Inspired by representation theory the PI plans to develop other tools to study varieties and their categories of coherent sheaves. For example, he plans to study actions of the Heisenberg Lie algebra on categories of sheaves on Hilbert schemes.The simplest example of a 3-manifold is the 3-dimensional world we live in. A little more complicated is what you obtain by cutting out a doughnut or perhaps an object with more holes. However, there are even more complicated 3-manifolds which are harder to describe. A fundamental problem in low-dimensional topology is how to tell whether two given 3-manifolds are "the same" or not ("the same" has a very precise mathematical definition in this case). A 3-manifold invariant is a tool for doing this. Their construction leads one to study other fields of mathematics and theoretical physics such as representation theory (the study of matrices) and algebraic geometry (the study of systems of polynomial equations). This rich connection to other fields is one of the attractive features of this problem.

利用代数几何和几何表示理论中的工具和技术构造三流形不变量。给定一个李代数，可以定义一个3-流形的数值不变量，称为Witten-Reshetikhin-Turaev不变量。将这些提升为3流形的同调不变量是一个开放的问题。这类似于拓扑空间的（奇异）同调提升欧拉特征的方式。在PI与Joel Kamnitzer的早期工作中，他们能够在李代数为sl(n)的情况下，对3球内的链路的补进行这样的处理，并且链路被基本表示标记。为了做到这一点，他们在类似旗子的变体上使用了某些（衍生的）相干束的类别，并希望将这种方法推广到其他3-流形。他们的结构是代数几何的，但使用了许多来自表示理论的技术。受表征理论的启发，PI计划开发其他工具来研究品种及其相干束的类别。例如，他计划研究海森堡李代数在希尔伯特格式的捆类上的作用。三维流形最简单的例子就是我们生活的三维世界。更复杂一点的是，你可以剪下一个甜甜圈，或者一个有更多洞的物体。然而，还有更复杂的3流形更难描述。低维拓扑中的一个基本问题是如何判断两个给定的3-流形是否“相同”（在这种情况下，“相同”有一个非常精确的数学定义）。3流形不变量是实现这一点的一个工具。它们的构建引导人们学习数学和理论物理的其他领域，如表示理论（研究矩阵）和代数几何（研究多项式方程系统）。这种与其他领域的丰富联系是这个问题的一个吸引人的特点。