Interactions between combinatorics, representation theory, and algebraic geometry

几何表示理论和奥比福德·希尔伯特模式这个项目希望用某些代数的表示来解释划分组合中的一些实验观察到的模式。这既证明了一些猜想，也为进一步发展这些猜想铺平了道路。想象一下平面上n个不可分辨的点的可能构形。当所有的点都是不同的时，这个空间看起来像2n维空间-，我们有2维空间，我们可以摆动n个点中的每一个。但如果其中两个点相撞，我们就不能再分辨出我们正在摆动的是哪个点，空间就会变得很糟糕，或者说是“奇异的”。N点的希尔伯特方案是一个密切相关的空间，它通过“记住这些点是如何在一起运行的”来修复这些奇点。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的划分--将n写成较小数字的和的方法。这种联系在理解点的Hilbet格式和证明关于划分的定理方面都被证明是有用的。Goettsche将这一结果推广到当我们的n个点在任意二维空间中游荡时，而不是仅仅在平面上，Nakajima和Grojnowski使用几何表示理论的思想给出了Goettsche结果的另一个证明。结果表明，如果我们看“Orbiold Hilbert格式”，我们需要n个点满足某些对称性，那么与Ellingsrud和Stromme发现的那些相似的模式是成立的。这些模式中的一些已经被几何证明，但一些仍然是猜想，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，中岛已经用不同的名字对Orbilold Hilbert格式的特例进行了广泛的研究，并导致了表示理论的巨大突破。这个项目将迈出第一步，将几何表示理论的故事扩展到中岛的例子之外。特别是，学生将证明类似于Nakajima和Grojnowski的Orbilold Hilbert方案的Heisenberg代数结果，在组合学、表示论和代数几何的不同领域之间架起桥梁。

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.