We prove that a nonlinear version of the Grothendieck-Katz conjecture (essentially in the form given by Ekhedahl, Shepherd-Barron and Taylor) is equivalent to the original Grothendieck-Katz conjecture together with a certain differential algebraic geometric/model-theoretic statement: a type over C(t) with “p-curvature 0 for almost all p” is nonorthogonal to the constants.

Certain solutions to Harper’s equation are discrete analogues of (and approximations to) the Hermite–Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscill...

In this paper, which is essentially a survey, we solve the global Cauchy problem on causal manifolds for hyperbolic systems of linear partial differential equations in the framework of hyperfunctions. Besides the classical Cauchy–Kowalevsky theorem, our proofs only use tools and ideas of purely algebraic and geometric nature from the microlocal theory of sheaves. Mathematics Subject Classificat...

Three essentially different but usually mixed notions of program invertibility are considered. Reversivity when each action has a full inverse. Reversibility when each action can be undone (right inverse). Retractability when erroneous result can be retracted down to error in data. Constructive logical, algebraic, programming and realization aspects are considered for those types of programs.

To any Hamiltonian action of a reductive algebraic group G on a smooth irreducible symplectic variety X we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. For cotangent bundles these invariants essentially coincide with those arising in the theory of equivarant embeddings. Using our approach we establish some properties of the latter invariants.

We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all Deligne–Faltings log structures (in particular simple normal crossings divisor) from the simplest case with characteristic generated by N (essentially the smooth divisor case).