Related papers: Two-dimensional Lagrangian singularities and bifur…

We introduce a new type of singularity for smooth maps from $4$-manifolds to surfaces, called an $M$-singularity, whose critical locus is a circle contained in a single fiber. We show that the monodromy around an $M$-singularity is a…

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on…

In this paper, we investigate the stability manifold of local models of orbifold quotients of elliptic curves. In particular, we describe a component of the stability manifold which maps as a covering space onto the universal unfolding…

We classify lagrangian fibrations on Nikulin orbifolds, a well studied class of singular irreducible holomorphic symplectic varieties, and prove they verify the SYZ conjecture.

This paper introduces pedal curves of spacelike frontals in the hyperbolic 2-space. We mainly investigate the singularities of these hyperbolic pedal curves of spacelike frontals for non-singular and singular dual curve germs. We then show…

We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a…

We study degenerations of non-simple principally polarized abelian surfaces to the boundary in the toroidal compactification of $\mathcal{A}_2$, and describe the degenerate abelian surfaces as well as the degenerate elliptic curves that…

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…

Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of…

A Lagrangian study of two-dimensional turbulence for two different geometries, a periodic and a confined circular geometry, is presented to investigate the influence of solid boundaries on the Lagrangian dynamics. It is found that the…

We define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of…

The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special…

Given a vector bundle with integrable connection $(V,\nabla)$ on a curve, if $V$ is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like…

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic…

Let f = 0 be an implicit singular plane curve. When deforming f = 0, inflections and vertex emerge from the singularities. In this papper, we classify the deformations of f = 0 with respect to the inflections and the vertices in the cases…

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider…

This paper is to investigate an elliptic fibration over $\mathbb{CP}^2$ arising from the Lagrange top from the viewpoint of complex algebraic geometry. The description of the discriminant locus of this elliptic fibration is given in detail.…

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle…

We give a mathematically precise statement of the SYZ conjecture between mirror space pairs and prove it for any toric Calabi-Yau manifold with the Gross Lagrangian fibration. To date, it is the first time we realize the SYZ proposal with…

We study the moduli space of four dimensional ordinary Lie algebras, and their versal deformations. Their classification is well known; our focus in this paper is on the deformations, which yield a picture of how the moduli space is…