Abstract:
Motivated by mirror symmetry, we consider the Lagrangian fibration $\R^4\to\R^2$ and Lagrangian maps $f:L\hookrightarrow \R^4\to \R^2$, exhibiting an unstable singularity, and study how the bifurcation locus of gradient lines, the integral curves of $\nabla f_x$, for $x\in B$, where $f_x(y)=f(y)-x\cdot y$, changes when $f$ is slightly perturbed. We consider the cases when $f$ is the germ of a fold, of a cusp and, particularly, of an elliptic umbilic.

Abstract:
We derive some important geometric identities for Lagrangian submanifolds immersed in a K\"ahler manifold and prove that there exists a canonical way to deform a Lagrangian submanifold by a parabolic flow through a family of Lagrangian submanifolds if the ambient space is a Ricci-flat Calabi-Yau manifold.

Abstract:
Given, in the Lagrangian torus fibration $R^4\to R^2$, a Lagrangian submanifold $L$, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies associated to the generating function of $L$, and it is provided with a holomorphic structure. Morse homology, however, is not defined along the caustic $C$ of $L$ or along the bifurcation locus $B$, where the family does not satisfy the Morse-Smale condition. The holomorphic structure is extended to the subset $C\cup B$, except cusps, yielding the so called quantum corrections to the mirror object.

Abstract:
This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

Abstract:
In this work we study the wavefront set of a solution u to Pu = f, where P is a pseudodifferential operator on a manifold with real-valued homogeneous principal symbol p, when the Hamilton vector field corresponding to p is radial on a Lagrangian submanifold contained in the characteristic set of P. The standard propagation of singularities theorem of Duistermaat-Hormander gives no information at the Lagrangian submanifold. By adapting the standard positive-commutator estimate proof of this theorem, we are able to conclude additional regularity at a point q in this radial set, assuming some regularity around this point. That is, the a priori assumption is either a weaker regularity assumption at q, or a regularity assumption near but not at q. Earlier results of Melrose and Vasy give a more global version of such analysis. Given some regularity assumptions around the Lagrangian submanifold, they obtain some regularity at the Lagrangian submanifold. This paper microlocalizes these results, assuming and concluding regularity only at a particular point of interest. We then proceed to prove an analogous result, useful in scattering theory, followed by analogous results in the context of Lagrangian regularity.

Abstract:
In this work we produce microlocal normal forms for pseudodifferential operators which have a Lagrangian submanifold of radial points. This answers natural questions about such operators and their associated classical dynamics. In a sequel, we will give a microlocal parametrix construction, as well as a construction of a microlocal Poisson operator, for such pseudodifferential operators.

Abstract:
A twin Lagrangian fibration, originally introduced by Yau and the first author, is roughly a geometric structure consisting of two Lagrangian fibrations whose fibers intersect with each other cleanly. In this paper, we show the existence of twin Lagrangian fibrations on certain symplectic manifolds whose mirrors are fibered by rigid analytic cycles. Using family Floer theory in the sense of Fukaya and Abouzaid, these twin Lagrangian fibrations are shown to be induced from fibrations by rigid analytic subvarieties on the mirror. As additional evidences, we discuss two simple applications of our constructions.

Abstract:
Ideas of Fukaya and Kontsevich-Soibelman suggest that one can use Strominger-Yau-Zaslow's geometric approach to mirror symmetry as a torus duality to construct the mirror of a symplectic manifold equipped with a Lagrangian torus fibration as a moduli space of simple objects of the Fukaya category supported on the fibres. In the absence of singular fibres, the construction of the mirror is explained in this framework, and, given a Lagrangian submanifold, a (twisted) coherent sheaf on the mirror is constructed.

Abstract:
We give geometric explanations and proofs of various mirror symmetry conjectures for $T^{n}$-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses fiberwise Fourier transformation together with base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex structures and complexified symplectic structures, $H^{p,q}$'s, Yukawa couplings; (ii) sl(2)xsl(2)-actions; (iii) holomorphic and symplectic automorphisms and (iv) A- and B-connections, supersymmetric A- and B-cycles, correlation functions. We also study (ii) for $T^{n}$-invariant hyperkahler manifolds.

Abstract:
We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.