Categorification and 3-Manifold Invariants in Algebraic Geometry

该建议是使用代数几何和几何表示理论的工具和技术来构造3-流形不变量。给定一个李代数，可以定义一个三维流形的数值不变量，称为Witten-Reshetikhin-Turaev不变量。如何将它们提升到三维流形的同调不变量是一个公开的问题。这类似于拓扑空间的（奇异）同调提升欧拉特征线的方式。在PI与Joel Kamnitzer的早期工作中，当李代数是sl（n）并且链接由基本表示标记时，他们能够对3-球面内的链接的补数这样做。为了做到这一点，他们使用了某些（派生）类别的连贯层旗状品种，并希望将这种方法推广到其他3-流形。他们的建设是代数几何，但使用了许多技术表示论。受表象理论的启发，PI计划开发其他工具来研究相干层的种类和类别。例如，他计划研究行动的海森堡李代数类别的层希尔伯特计划。最简单的例子，一个3流形是3维世界，我们生活在。稍微复杂一点的是，你通过切割一个甜甜圈或一个有更多洞的物体来获得什么。然而，还有更复杂的三维流形，更难描述。低维拓扑学中的一个基本问题是如何判断两个给定的三维流形是否“相同”（“相同”在这种情况下有非常精确的数学定义）。一个3-流形不变量是一个工具来做到这一点。他们的建设导致一个研究其他领域的数学和理论物理，如表示论（研究矩阵）和代数几何（研究系统的多项式方程）。这种与其他领域的丰富联系是这个问题的吸引人的特点之一。

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.