Related papers: 5-th order differential equations related to Calab…
Using model theory and differential algebra, we give necessary conditions for algebraic ordinary differential equations to have a complex Pfaffian solution on some complex domain. These tools also allow us to give many examples of algebraic…
We study the derived equivalence of Calabi-Yau algebras and show that, for two derived Morita equivalent algebras, if one is Calabi-Yau, then so is the other. Keywords: Derived equivalence, Calabi-Yau algebra
In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions…
We prove that the bounded derived category of the lattice of order ideals of the product of two ordered chains is fractionally Calabi-Yau. We also show that these lattices are derived equivalent to higher Auslander algebras of type A. The…
Birational Calabi-Yau threefolds in the same deformation family provide a `weak' counterexample to the global Torelli problem, as long as they are not isomorphic. In this paper, it is shown that deformations of certain desingularized…
We introduce relative noncommutative Calabi-Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a…
As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to…
We consider algebras defined from quivers with relations that are k-th order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show…
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…
We study exceptional quotient singularities. In particular, we prove an exceptionality criterion in terms of the $\alpha$-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient…
Previously we constructed Calabi-Yau threefolds by a differential-geometric gluing method using Fano threefolds with their smooth anticanonical $K3$ divisors (New York J. Math. 20: 1-33, 2014). In this paper, we further consider the…
We perform a Kaluza-Klein reduction of IIB supergravity including purely gravitational $\alpha '^3$-corrections on a Calabi-Yau threefold, and perform the orientifold projection accounting for the presence of $O3/O7$-planes. We consider…
This is a report of my talk at MFO after Cetraro this July consisting of more speculations, rather than definite results, on automorphisms of strict Calabi-Yau threefolds in the view of the question posed in my ICM report after Dinh, Sibony…
This note is a report on the observation that the Enriques-Fano threefolds with terminal cyclic quotient singularities admit Calabi-Yau threefolds as their double coverings. We calculate the invariants of those Calabi-Yau threefolds when…
We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi-Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories,…
Using recent results on eleven-dimensional superspace, we compute the contribution of fivebrane instantons with four fermionic zeromodes in M-theory compactifications on Calabi-Yau fourfolds with flux. We find that no superpotential is…
In this paper we study various difference equations related to Jacobi-type pencils. By a Jacobi-type pencil one means the following pencil: $J_5 - \lambda J_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
The method of parameter variation for linear differential equations is extended to classes of second order nonlinear differential equations. This allows to reduce the latter to first order differential equations. Known classical equations…
The fractional Feynman-Kac equations describe the distribution of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the fractional…