Abstract by C.J. Bott
Mirror Symmetry for K3 Surfaces With Non-symplectic Automorphism
Mirror symmetry is the phenomenon originally discovered by physicists that Calabi-Yau manifolds come in dual pairs, each of which produces the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. There are several constructions in different situations for constructing the mirror dual of a Calabi-Yau manifold. It is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?
We consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry, the first inspired by the Landau-Ginzburg/Calabi-Yau correspondence, and the second more classical. In particular, for certain K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani-Boissière-Sarti and Comparin-Lyon-Priddis-Suggs show that they agree when n is prime. We will discuss new techniques needed to solve the problem when n is composite.