Symmetry in classical enumerative geometry

In this talk we’ll discuss an equivariant principle of conservation of number, proven using methods from equivariant homotopy theory. It roughly states that in the presence of symmetry, not only the number of solutions is conserved, but their symmetries are as well. For instance when a cubic surface is defined by a symmetric polynomial, its 27 lines always carry the same S4 action. We apply this idea in joint work with C. Bethea to compute bitangents to smooth plane quartics with nontrivial automorphism groups, where we see that homotopical techniques directly reveal patterns which are not obvious from a classical moduli perspective. We will also discuss work with S. Raman, in which we initiate a study of Galois groups of symmetric enumerative problems, leveraging tools from Hodge theory and computational numerical analysis