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Singular Lefschetz pencils
We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4–manifold equipped with a “near-symplectic” structure (ie, a closed 2–form which is symplectic outside a
Flat surfaces and stability structures
We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface Xk
Families of K3 surfaces
We use automorphic forms to prove that a compact family of Kaehler K3 surfaces with constant Picard number is isotrivial.
Variation of geometric invariant theory quotients and derived categories
We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions
Mirror symmetry for weighted projective planes and their noncommutative deformations
We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror
Dynamical systems and categories
We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and
Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models
In this paper we prove the smoothness of the moduli space of Landau-Ginzburg models. We formulate and prove a Tian-Todorov theorem for the deformations of Landau-Ginzburg models, develop the
Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves
Abstract. We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely
Hodge theoretic aspects of mirror symmetry
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of