ENUMERATIVE GEOMETRY: SSM CLASSES OF MULTISINGULARITIES AND CURVE COUNTING

In this thesis, we study two separate frameworks linking representation theory and (enumerative) algebraic geometry. In the first setting, we study the geometry of double point loci of maps $F:M\to N$ of complex manifolds through the lens of Segre-Schwartz-MacPherson (SSM) classes. Classical double point formulas express the geometry of the double point locus of $F$ in terms of global invariants of source and target spaces, as well as $F$. In the algebro-geometric setting, such formulas were discovered by Fulton and Laksov. We extend these results by computing a one-parameter cohomological deformation of the double point formula given by the SSM class, which recover and generalize Fulton and Laksov's result in a large cohomological degree range. Our approach uses interpolation techniques for SSM-Thom polynomials of multisingularities, recently developed by Koncki, Nekarda, Ohmoto and Rim\'anyi. We also compute SSM-Thom polynomials for the singularities $A_0$ and $A_1$ in the same range. As an application, use a theorem of Aluffi and Ohmoto to show how the deformed formulas yield refined geometric information about those singularity loci, including constraints on when such loci can arise as complete intersections. This work is based on the author's recent paper, \cite{ReeseSSM}. In the second chapter, we study quiver gauge theories; mathematical descriptions of 3d, $\mathcal N=4$ quantum field theories determined by combinatorial data called a quiver. We introduce a new operation called the \textit{slant sum} of a pair of quiver gauge theories, a gluing-type operation on the underlying quivers. Under some mild assumptions, we relate torus fixed points on the corresponding Higgs branches, which are Nakajima quiver varieties. The main result of this section concerns the interaction of the new slant sum operation with a classical object of study in K-theoretic enumerative geometry, the quasimap vertex function, an equivariant generating function for counts of quasimaps to a variety. We prove a branching rule relating vertex functions to a quiver variety before and after a slant sum, and deduce a number of factorization corollaries. Our construction and branching rule are motivated by a conjecture we introduce for the factorization of the vertex functions of zero-dimensional quiver varieties, whose proof can be approached inductively using our branching rule. In special cases, it also shows that vertex functions can be written as sums over reverse plane partitions, even beyond ADE type. Vertex functions and their factorizations are now understood to play an important role in the study of 3d mirror symmetry, a predicted duality among Higgs and Coulomb branches arising from 3d, $\mathcal N=4$ quantum field theories. We explain how our branching rule interacts with the predictions of 3d mirror symmetry. This work is based on the author's joint work with Hunter Dinkins and Vasily Krylov, \cite{ReeseSS}.