Interactions between combinatorics, representation theory, and algebraic geometry

几何表示理论和orbifold希尔伯特schemes这个项目希望解释一些实验观察到的模式在组合的分区在某些代数的表示。这既可以证明一些假设，也可以引导进一步发展它们。想象一下平面上n个不可区分的点的可能构型。当所有的点都是不同的，这个空间看起来像2n维空间--我们有两个维度，我们可以摆动n个点中的每一个。但是如果两个点碰撞，我们就不能再分辨出我们是在扭动哪一点，空间就会变得不好，或者说是“奇异”。希尔伯特方案的n个点是一个密切相关的空间，固定这些奇异性通过“记住点如何运行在一起”。这是一个复杂的空间，但Ellingsrud和Stromme证明了它的形状或“拓扑”可以用更简单的组合对象来描述，称为n的分区-将n写成较小数字之和的方式。这种联系在理解点的希尔伯特方案和证明关于划分的定理方面都被证明是有用的。Goettsche将这个结果扩展到当我们的n个点在任意二维空间中徘徊时，而不仅仅是平面，Nakajima和Grojnowski利用几何表示理论的思想对Goettsche的结果给出了另一个证明。如果我们看一下“Orbifold Hilbert schemes”，我们需要n个点满足某些对称性。这些模式中的一些已经被几何证明，但有些仍然是几何图形，尽管它们看起来都可以从几何表示理论中得到证明。事实上，Orbifold希尔伯特方案的特殊情况已经被Nakajima以不同的名字广泛研究，并导致了表示论的巨大突破。这个项目将使几何表示理论的故事超出Nakajima的例子的第一步。特别是，学生将证明模拟Nakajima和Grojnowski的海森堡代数结果的Orbifold希尔伯特计划，建立桥梁之间的组合学，表示论和代数几何的独立领域。

Geometric representation theory and orbifold Hilbert SchemesThis project hopes to explain some experimentally observed patterns in the combinatorics of partitions in terms of representations of certain algebras. This would both prove some conjectures, and lead the path forward to developing them further.Imagine the possible configurations of n indistinguishable points in the plane. When all the points are distinct, this space looks like 2n dimensional space -- we have two dimensions we can wiggle each of the n points. But if two of the points collide, we can no longer tell which point we're wiggling away, and the space becomes badly behaved, or "singular" there. The Hilbert scheme of n points is a closely related space that fixes these singularities by "remembering how the points ran together". It is a complicated space, but Ellingsrud and Stromme proved that it's shape or "topology" can be described in terms of simpler, combinatorial objects known as the partitions of n -- the ways of writing n as a sum of smaller numbers. This connection has proven useful both in understanding the Hilbet scheme of points, and in proving theorems about partitions.Goettsche extended this result to when our n points are wandering around an arbitrary two dimensional space, instead of just the plane, and Nakajima and Grojnowski gave another proof of Goettsche's result using ideas from geometric representation theory.It turns out that similar patterns to those found by Ellingsrud and Stromme hold if we look at "Orbifold Hilbert schemes", where we require the n points to satisfy certain symmetries. Some of these patterns have been proven geometrically, but some remain conjectures, and though all of them look like they could have proofs coming from geometric representation theory. In fact, special cases of the Orbifold Hilbert schemes have already been studied extensively by Nakajima under a different name, and led to huge breakthroughs in representation theory. This project will make the first steps in extending the Geometric Representation Theory story beyond Nakajima's examples. In particular, the student will prove an analog of Nakajima and Grojnowski's Heisenberg algebra result for Orbifold Hilbert schemes, building bridges between the separate areas of combinatorics, representation theory, and algebraic geometry.