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

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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.