Interactions between combinatorics, representation theory, and algebraic geometry

几何表示理论和轨道希尔伯特图式本项目希望用某些代数的表示来解释在分区组合学中一些实验观察到的模式。这既证明了一些猜想，又为进一步发展这些猜想开辟了道路。想象平面上n个不可区分点的可能构型。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，我们可以摆动这n个点。但如果两个点碰撞，我们就无法分辨我们摆动的是哪一点，空间就会变得不稳定，或者说“奇异”。n个点的希尔伯特格式是一个紧密相关的空间，通过“记住这些点是如何一起运行的”来固定这些奇点。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，这些组合对象被称为n的分区——将n写成小数字的和的方式。这种联系已被证明对理解Hilbet点格式和证明关于分区的定理都是有用的。Goettsche将这个结果扩展到n个点在任意二维空间中，而不是在平面上，Nakajima和Grojnowski用几何表示理论的思想给出了Goettsche结果的另一个证明。如果我们看一下“奥比德希尔伯特方案”，我们会发现与Ellingsrud和Stromme发现的相似的模式，我们需要n个点来满足一定的对称性。其中一些模式已经被几何证明了，但有些仍然是猜测，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，Nakajima已经以不同的名义对Orbifold Hilbert格式的特殊情况进行了广泛的研究，并导致了表征理论的巨大突破。这个项目将迈出第一步，将几何表征理论的故事扩展到中岛的例子之外。特别是，学生将证明Nakajima和Grojnowski的海森堡代数结果在轨道希尔伯特方案中的类比，在组合学，表示理论和代数几何的不同领域之间建立桥梁。

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.