Equivariant cohomology classes in quiver theory and statistical mechanics, and, a more geometric foundation of intersection theory

我的建议的一个共同特点是使用代数几何退化来研究一个有趣的不可约品种的许多组件，它打破了限制。这样的退化通常是非常破坏性的几何，但原来的和limitingschemes可能会定义相同的（等变）上同调类在某些环境空间。我打算研究的退化是：（1）箭图表示的模空间，其中A型的情形在过去三年中已被很好地理解，但D、E型是完全开放的;（2）李代数中的幂零轨道，退化到其上三角部分的正规锥。上三角形部分的同源性带有一个（施普林格）表示相应的Weyl群;正常锥我研究有更多的组件，并在exampleI可以计算到目前为止，进行了更丰富的代数表示;（3）射影空间的子簇到射影空间的有限映射的约化概型，其中，映射的次数携带着在极限子概型的非约化结构中更常见的信息。如果我们用一个多项式方程（如xy = zw）替换它的一个单项式（如xy = 0），通常可以使溶液的轨迹保持相同的体积，即使它破碎成碎片。 其中xy=zw有有趣的非平坦几何，但不能分解成块，xy= 0有有趣的离散结构（它有两个部分，x=0和y=0，它们彼此相交），即使每一部分是平坦的。在一般的设置中，我们有许多多项式条件开始，每一个都退化为”一个只包含一些原始单项式的简单多项式;在我的工作中，我研究了周围空间是矩阵空间，原始方程是自然矩阵条件，如“平方为零的矩阵”的退化问题。困难在于找到保持尺寸和体积的简并，然后研究所得到的方程，以理解简并轨迹因子的简单片段。由此产生的discrete结构，然后极大的兴趣combinatorially，和sheds轻的（拓扑重要的）体积的原始轨迹。

A common feature of my proposals is the use of algebro-geometricdegeneration to study an interesting irreducible variety in terms ofthe many components it breaks into in a limit. Such degenerations are usually very destructive of the geometry, but the original and limitingschemes may define the same (equivariant) cohomology class in someambient space. The degenerations I intend to study are (1) of modulispaces of representations of quivers, where the type A case has becomewell understood in the last three years but types D,E are totally open;(2) of nilpotent orbits in Lie algebras, to the normal cones of theirupper triangular part. The homology of the upper triangular part carries a (Springer) representation of the corresponding Weyl group;the normal cone I study has more components and, in the exampleI can compute so far, carries a representation of a richer algebra;(3) of subvarieties of projective space to reduced schemes withfinite maps to projective space, where the degree of the map carriesthe information more usually seen in the nonreduced structure of the limit subscheme.If we replace a polynomial equation, like xy = zw, by one of itsmonomials, like xy = 0, the locus of solutions can often be made toretain the same volume even as it breaks into pieces. Where xy=zw hasinteresting nonflat geometry, but cannot be factored into pieces, xy=0has interesting discrete structure (it has two pieces, x=0 and y=0,which intersect one another) even though each piece is flat.In the general setup, we have many polynomial conditions to begin with, and each is degenerated" to a simpler polynomial containing only some ofthe original monomials; a complicated condition guarantees that thedimension and volume of the solution set remains constant.In my work I study these degenerations where the ambient space is aspace of (lists of) matrices, and the original equations are natural matrix conditions like "matrices whose square is zero". The difficulty comes in finding degenerations that preserve the dimension and volume,and then studying the resulting equations to understand the simplerpieces into which the degenerate locus factors. The resulting discretestructure is then of great interest combinatorially, and sheds lighton the (topologically important) volume of the original locus.