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Suppose we are given a del Pezzo surface and a full exceptional collection in its derived category. One may associate a toric system to this collection. Such a toric system defines a toric variety, which seems closely related to a toric degeneration of the original del Pezzo surface. In this talk we review basic notions related to toric systems, and present examples illustrating how these information can be exploited to study toric degenerations of del Pezzo surfaces.

We recall the notion of a twisted Jacobian ring of a potential with a diagonal group action, which appear naturally in TQFT and mirror symmetry. Then we discuss how we can see the structure in Lagrangian Floer theory which encodes data of holomorphic discs bounded on Lagrangian submanifolds in symplectic manifolds. We will provide an example which features all properties of twisted Jacobian rings. This is a joint work with Cheol-Hyun Cho.

Consider a nonsingular projective curve embedded in projective space by the complete linear system of a very ample line bundle with sufficiently large degree. It is a classical result of Castelnuovo, Mumford, and many others that this curve is projectively normal and the defining ideal is generated by quadrics. Green realized that the classical questions on defining equations should be generalized to higher syzygies, and proved his famous (2g+1+p)-theorem. In this talk, I show that secant varieties of the curve are arithmetically Cohen-Macaulay and satisfies the property N_{k+2,p}. This result was conjectured by Sidman-Vermeire, and it can be regarded as a generalization of Green's theorem. I also prove that those secant varieties have mild singularities naturally appearing in birational geometry. This talk is based on joint work with Lawrence Ein and Wenbo Niu.

Given a Lagrangian in a symplectic manifold, one can consider its Maurer-Cartan deformation which produces a local chart of the mirror that encodes mirror geometry near this Lagrangian. Construction applies to the union of Lagrangian spheres in a certain open symplectic manifold, and produces noncommutative resolutions of well-known algebraic singularities, which are in the form of quivers with potentials. In this talk, I will examine such a construction in different dimensions, and explain how quivers can be used to effectively compare mirror geometries. In particular, in dimension 3, one can find an interesting symplectic operation which is mirror to Atiyah flop on the mirror side.