Calabi-Yau differential equations of degree 2 and 3 and Yifan Yang's pullback

This short note is an extended abstract of a talk given at the conference "Komplexe Analysis" at the Mathematisches Forschungsinstitut Oberwolfach in September 2012. We explained some recent results about the existence of rational curves on Calabi-Yau threefolds as well as a curvature approach to the non hyperbolicity of such manifolds.

We study triples of graded rings defined over the deformation spaces for certain one-parameter families of Calabi-Yau threefolds. These rings are analogues of the rings of modular forms, quasi-modular forms and almost-holomorphic modular forms. We also discuss some of their applications in solving the holomorphic anomaly equations.

With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and non-compact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have been useful in the interaction between the physics and the mathematics, especially in string and gauge theories. A skein which runs through this review will be algorithmic and computational algebraic geometry and how, implementing its princip...

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-K\"ahler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Amp\`ere equation having a unique solution. In this way we prove that for every total space $M^4$ of an orientable $T^2$-bundle over $\mathbb{T}^2$ endowed with an invariant almost-K\"ahler structure the Calabi-Yau problem has a solution for every no...

We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.

The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward the following hypothesis. Any integrable equation of the type under consideration admits an infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of a finite-field system is formalized as fini...

This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.

We study tuples of matrices with rigidity index two in $\Sp_4(\mathbb{C})$, which are potentially induced by differential operators of Calabi-Yau type. The constructions of those monodromy tuples via algebraic operations and middle convolutions and the related constructions on the level differential operators lead to previously known and new examples.

We propose a new method for solution of the integrability problem for evolutionary differential-difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is integrable or not, and to advance in classification of integrable equations. In this paper we define and develop symbolic representation for the difference polynomial ring, difference operators and formal series. In order to formulate necess...

This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence problem is to determine conditions on two differential operators such that there exists a fiber-preserving transformations mapping one to the other according to ga...