Symmetry in Enumerative Geometry

Classical enumerative geometry studies problems of counting geometric objects satisfying specific conditions. Such counts are typically expressed as integers, and can often be obtained using topological invariants like the Euler number. In recent years, a number of refinements of enumerative invariants have emerged, producing new counts and revealing additional structure in classical counts. In this lecture, I will describe an approach to understanding symmetries in enumerative geometry when a finite group acts on the set of solutions to an enumerative problem. This naturally leads to a theory of equivariantly enriched enumerative geometry. I will present several examples of equivariant enumerative results, including a count of rational cubic curves and bitangents to symmetric quartic curves, joint with a number of authors including K. Wickelgren, T. Brazelton, and C. Ravi.