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种有趣的不可约种类，就它在一个极限中分解成的许多分量而言。这种简并通常对几何具有很大的破坏性，但原始方案和限制方案可能在某个环境空间中定义相同的(等变)上同调类。我打算研究的退化是(1)箭图表示的模空间，其中A型情况在过去三年中已被很好地理解，但D型，E型是完全开放的；(2)李代数中的幂零轨道到其上三角部分的法锥的退化。上三角部分的同调带有相应Weyl群的(Springer)表示；我研究的法锥有更多的分量，并且，在我到目前为止所能计算的例子中，带有更丰富的代数的表示；(3)射影空间的子簇到有限映射到射影空间的约化格式，其中映射的次数携带了在极限子格式的非约化结构中更常见的信息。如果我们将多项式方程，如xy=zw，替换为它的单项式，如xy=0，解的轨迹经常可以使它保持相同的体积，即使它分裂成碎片。其中xy=zw具有有趣的非平坦几何，但不能分解成片段，xy=0具有有趣的离散结构(它有两个片段，x=0和y=0，它们彼此相交)，即使每一片段都是平的。在一般设置中，我们有许多多项式条件，并且每个条件都退化为只包含一些原始单项式的更简单的多项式；复杂的条件保证解集的维度和体积保持不变。在我的工作中，我研究了这些退化，其中环境空间是(列表)矩阵的空间，原始方程是自然的矩阵条件，如“平方为零的矩阵”。困难在于找到保持维度和体积不变的简并，然后研究得到的方程，以理解简并轨迹因数所成的更简单的部分。由此产生的离散结构在组合上是非常有意义的，并减轻了原始轨迹的体积(在拓扑上很重要)。

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.