Categorification and 3-Manifold Invariants in Algebraic Geometry

建议使用代数几何和几何表示理论中的工具和技巧来构造三维流形不变量。给定一个李代数，我们可以定义一个三维流形的数值不变量，称为Witten-Reshetikhin-Turaev不变量。将它们提升到三维流形的同调不变量是一个悬而未决的问题。这类似于拓扑空间的(奇异)同调提升欧拉特征的方式。在PI和Joel Kamnitzer的早期工作中，当李代数是sl(N)时，他们能够对3-球面内的链环的补集做这项工作，并且用基本表示来标记链环。为了做到这一点，他们使用了旗型簇上的某些(派生的)相干层范畴，希望将这种方法推广到其他3-流形上。它们的构造是代数几何的，但使用了许多表示论的技术。在表象理论的启发下，PI计划开发其他工具来研究相干滑轮的品种及其类别。例如，他计划研究Heisenberg李代数在Hilbert模式上的层范畴上的作用。3-流形的最简单的例子是我们生活的3维世界。稍微复杂一点的是，你通过切出一个甜甜圈或一个有更多洞的物体所获得的东西。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区分两个给定的三维流形是否“相同”(在这种情况下，“相同”有一个非常精确的数学定义)。3-流形不变量就是这样做的一个工具。它们的构造引导人们研究数学和理论物理的其他领域，如表示论(研究矩阵)和代数几何(研究多项式方程组)。这种与其他领域的丰富联系是这个问题的吸引人的特征之一。

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.