In this expository paper, we discuss how Fourier–Mukai-type transformations, which we call SYZ mirror transformations, can be applied to provide a geometric understanding of the mirror symmetry phenomena for semi-fiat Calabi–Yau manifolds and toric Fano manifolds. We also speculate the possible applications of these transformations to other more general settings.

An effective family of spectral curves appearing in Hitchin fibrations is determined. Using this family the moduli spaces of stable Higgs bundles on an algebraic curve are embedded into the Sato Grassmannian. We show that the Hitchin integrable system, the natural algebraically completely integrable Hamiltonian system defined on the Higgs moduli space, coincides with the KP equations. It is shown that the Serre duality on these moduli spaces corresponds to the formal adjoint of pseudo-differential operators acting on the Grassmannian. From this fact we then identify the Hitchin integrable system on the moduli space of $Sp_{2m}$-Higgs bundles in terms of a reduction of the KP equations. We also show that the dual Abelian fibration (the SYZ mirror dual) to the $Sp_{2m}$-Higgs moduli space is constructed by taking the symplectic quotient of a Lie algebra action on the moduli space of $GL$-Higgs bundles.

We study certain differential rings over the moduli space of Calabi–Yau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasi-modular forms due to Kaneko–Zagier[23].

This is an expository article on the recent studies [27, 28, 45, 46, 23] of Ruan's crepant resolution/flop conjecture [62, 63] and its possible relations to the $K$-theory integral structure [45, 46, 49] in quantum cohomology.

We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semi-stable variety of form $xy = 0$. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramified stable maps and stable relative maps. As an application, we obtain a modular desingularization of the main component of Kontsevich's moduli space of elliptic stable maps to a projective space.

This is a review of the authors' recent results on an integrable structure of the melting crystal model with external potentials. The partition function of this model is a sum over all plane partitions (3D Young diagrams). By the method of transfer matrices, this sum turns into a sum over ordinary partitions (Young diagrams), which may be thought of as a model of $q$-deformed random partitions. This model can be further translated to the language of a complex fermion system. A fermionic realization of the quantum torus Lie algebra is shown to underlie therein. With the aid of hidden symmetry of this Lie algebra, the partition function of the melting crystal model turns out to coincide, up to a simple factor, with a tau function of the 1D Toda hierarchy. Some related issues on 4D and 5D supersymmetric Yang–Mills theories, topological strings and the 2D Toda hierarchy are briefly discussed.

We study topological A-type string on an arbitrary two dimensional target space. Using the Virasoro constraints, proven by A. Okounkov and R. Pandharipande, we find an explicit formula for the partition function. The target space field theory reproducing this partition function is proposed. This field theory has infinite set of deformations which are overlooked by the standard definition of the topological string. We also discuss the relations to the multi-trace deformations of gauge theories, and make contact with quantum integrable systems. In addition, the target space theory can be in turn coupled to gravity, thereby realizing the topological string version of M. Green's "worldsheets for worldsheets" idea.

Gromov–Witten theory is used to define an enumerative geometry of curves in Calabi–Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov–Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi–Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise.

Two main examples are considered: the local Calabi–Yau $\mathbb{P}^2$ with normal bundle $\oplus_{i=1}^{3} \mathcal{O} (-1)$ and the compact Calabi–Yau hypersurface $X_7 \subset \mathbb{P}^6$. In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Strömme.

We show that the limit, by rescaling, of the 'new supersymmetric index' attached to the Fourier–Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a $E_1$-degeneration property for it.

$\mathcal{A}$ is a set of $N$ vectors in $\mathbb{Z}^{N-2}$ situated in a hyperplane not through 0 and spanning $\mathbb{Z}^{N-2}$ over $\mathbb{Z}$. Gulotta's algorithm [4] constructs from $\mathcal{A}$ a dimer model. $\mathcal{A}$ theorem in [6] states that the principal $\mathcal{A}$-determinant equals the determinant of (a suitable form of) the Kasteleyn matrix of that dimer model. In the present note we translate Gulotta's pictorial description of the algorithm into matrix operations. As a result one obtains an algorithm for computing the principal $\mathcal{A}$-determinant, which is much faster than the algorithm in [5].

We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system of weights and the other is defined via the signature of the same regular system of weights.

The main result in this paper is that if a regular system of weights is of dual type then these two weighted projective lines have equivalent abelian categories of coherent sheaves. As a corollary, we can show that the triangulated categories of the graded singularity associated to a regular system of weights has a full exceptional collection, which is expected from homological mirror symmetries.

The main theorem of this paper will be generalized to more general one, to the case when a regular system of weights is of genus zero, which will be given in [5]. Since we need more detailed study of regular systems of weights and some knowledge of algebraic geometry of Deligne–Mumford stacks there, the author write a part of the result in this paper to which another simple proof based on the idea by Geigle–Lenzing [2] can be applied.

The notion of limit stability on Calabi–Yau 3-folds is introduced by the author to construct an approximation of Bridgeland–Douglas stability conditions at the large volume limit. It has also turned out that the wall-crossing phenomena of limit stable objects seem relevant to the rationality conjecture of the generating functions of Pandharipande–Thomas invariants. In this article, we shall make it clear how wallcrossing formula of the counting invariants of limit stable objects solves the above conjecture.