Ailsa Keating - Research

I'm interested in symplectic topology and homological mirror symmetry, and connections to singularity theory, algebraic geometry, and geometric group theory.

Preprints

Homological mirror symmetry for projective K3 surfaces, with Paul Hacking

We prove the homological mirror symmetry conjecture of Kontsevich for K3 surfaces in the following form: The Fukaya category of a projective K3 surface is equivalent to the derived category of coherent sheaves on the mirror, which is a K3 surface of Picard rank 19 over the field C((q)) of formal Laurent series. This builds on prior work of Seidel, who proved the theorem in the case of the quartic surface, Sheridan, Lekili--Ueda, and Ganatra--Pardon--Shende.

A universal mirror to (P^2,Ω) as a birational object, with Abigail Ward

We study homological mirror symmetry for (P^2,Ω) viewed as an object of birational geometry, with Ω the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme U_univ and a Weinstein manifold M_univ, both of infinite type; and we prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of U_univ are given by a natural discrete subgroup of Bir(P^2,±Ω); and that all of these automorphisms are mirror to symplectomorphisms of M_univ. We conclude with some applications.

Articles