Jenia Tevelev: An exercise in homological mirror symmetry

Calculating the endomorphism ring of a vector bundle in algebraic geometry can be tricky because there is no natural basis. By contrast, the endomorphism ring of a Lagrangian in symplectic geometry often does have a natural basis given by its self-intersection points. Mirror symmetry provides a way to reduce the former problem to the latter. As an example, we first work out the multiplication in the Kalck–Karmazyn algebra (the endomorphism algebra of a Kawamata vector bundle on a two-dimensional cyclic quotient singularity). Then we use the Fukaya-Oh-Ohta-Ono formalism to compute its deformation to the matrix algebra discovered by Kawamata, or more generally to a hereditary algebra. Joint work with Yanki Lekili.