Symbolic Computation Group

The main part of this talk will be devoted to modern generalizations of factorization to the cases of systems of linear partial differential equations and their relation with explicit solvability of nonlinear ordinary and partial differential equations based on some constructions from the theory of abelian categories.

We will briefly review classical algebraic theory of factorization of linear ordinary differential equations (G. Frobenius, E. Landau, A.Loewy, O.Ore, 1880-1930) and their properties (uniqueness etc). We give links for this problem to modern algorithms for factorization of linear ordinary differential equations (singularities etc. along with links to differential Galois theory).

We will then discuss decomposition of non-linear ordinary differential equations: Sosnin's algorithm (2001) and the general problem of decomposition. Reduction to the problem of factorization of linear ordinary differential equations over transcendental extensions.

This will be followed by a discussion on actorization of non-linear partial differential operators. Methods include ``naive'' and ``generalized'' factorization of linear partial differential equations, Laplace transformations for linear equations and Darboux integrability of nonlinear partial differential equations. Description in terms of special ideals in the ring of linear partial differential operators. Results on generalized factorization for more than two independent variables (after an idea by U.Dini, 1902) are also presented.

Finally we look at Serre-Grothendiek factor categories and the general definition of factorization of arbitrary systems of linear partial differential equations. This will include algorithmic problems and complexity of the factors.