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变量，根据它在极限中分解成的许多分量。这种退化通常对几何是非常有害的，但是原始方案和极限方案可以在某些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)颤振表示的模空间，其中A型情况在过去三年中已经得到了很好的理解，但D，E型是完全开放的；(2)李代数中的幂零轨道，到它们的上三角部分的正规锥。上三角部分的同调带有对应Weyl群的（施普林格）表示；我研究的正常锥有更多的组成部分，在我目前能计算的例子中，它代表了更丰富的代数；(3)将射影空间的子变体转化为具有有限映射到射影空间的约简格式，其中映射的度携带了在极限子格式的非约简结构中更常见的信息。如果我们把一个多项式方程，比如xy = zw，换成它的一个单项式方程，比如xy = 0，那么即使解的轨迹被分解成碎片，它的体积通常也能保持不变。xy=zw具有有趣的非平面几何，但不能分解成几个部分，而xy=0具有有趣的离散结构（它有两个部分，x=0和y=0，它们彼此相交），尽管每个部分都是平面的。在一般设置中，我们有许多多项式条件开始，每个条件退化为一个只包含一些原始单项式的更简单的多项式；一个复杂的条件保证了解集的尺寸和体积保持不变。在我的工作中，我研究这些退化，其中环境空间是矩阵的空间（列表），原始方程是自然矩阵条件，如“矩阵的平方为零”。难点在于找到保持维度和体积不变的退化，然后研究得到的方程，以理解退化轨迹因子分解成的更简单的部分。由此产生的离散结构在组合上是非常有趣的，并且揭示了原始轨迹的（拓扑上重要的）体积。

A common feature of my proposals is the use of algebro-geometric degeneration to study an interesting irreducible variety in terms of the many components it breaks into in a limit. Such degenerations are usually very destructive of the geometry, but the original and limiting schemes may define the same (equivariant) cohomology class in some ambient space. The degenerations I intend to study are (1) of moduli spaces of representations of quivers, where the type A case has become well 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 their upper 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 example I can compute so far, carries a representation of a richer algebra; (3) of subvarieties of projective space to reduced schemes with finite maps to projective space, where the degree of the map carries the information more usually seen in the nonreduced structure of the limit subscheme.If we replace a polynomial equation, like xy = zw, by one of its monomials, like xy = 0, the locus of solutions can often be made to retain the same volume even as it breaks into pieces. Where xy=zw has interesting nonflat geometry, but cannot be factored into pieces, xy=0 has 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 of the original monomials; a complicated condition guarantees that the dimension and volume of the solution set remains constant. In my work I study these degenerations where the ambient space is a space 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 simpler pieces into which the degenerate locus factors. The resulting discrete structure is then of great interest combinatorially, and sheds light on the (topologically important) volume of the original locus.