We consider mirror symmetry for a log Calabi--Yau pair (X,D), where X is a Fano variety and D is an anticanonical divisor of X. The mirror of the pair (X,D) is a Landau--Ginzburg model (X^\vee, W), where W is a function called the superpotential. Following the mirror constructions in the Gross--Sibert program, W can be described in terms of Gromov--Witten invariants of (X,D). I will explain a relative mirror theorem for the pair (X,D). As applications, I will explain how to compute the superpotential W, the classical period of W and more.

I give an overview of a project with Michael Wemyss to classify simple 3-fold flops. This amounts to understanding which smooth 3-fold neighbourhoods C inside X of a rational curve C with normal bundle O(-3)+O(1) can be contracted. (The other possible flopping normal bundles are the Atiyah flop and Reid's Pagoda flops.)

As X is a manifold, one can describe the situation by glueing together patches, and in fact it is enough to glue together two copies of the affine space C^3 by a simple formula. However, the formula has infinitely many free parameters, and most choices describe neighbourhoods that do not contract. Even just finding examples of good glue functions seems to be needle-in-a-haystack. Instead, we translate the problem to one of classifying noncommutative germs f(x,y), or equivalently complete local Jacobian algebras up to isomorphism, as the putative contraction algebras of C in X. In this context, the conditions to make the right choices of parameters seem to be much more controllable.

Working with such germs feels much like a noncommutative version of classical singularity theory of function germs in the style of Arnold (types ADE and all that), and we can solve enough of this problem to construct all flops and to provide the classification. The resulting ADE classification presents Atiyah-Reid as Type A, an infinite discrete collection for Type D, and then families with moduli for the four (!) classes of Type E.

It is notoriously difficult to construct stable bundles with given topological invariants. This difficultly is partially because we have few general constructions of stable vector bundles. Recent work has focused on a conjectural construction called syzygy bundles​ which are bundles arising as the kernel of the evaluation map on global sections. In this talk, we show syzygy bundles associated to globally-generated line bundles are stable---confirming this conjectural construction.