Classification of Fuchsian Systems and Their Connection Problem

UTMS 2008 – 29 October 28 , 2008 Classification of Fuchsian systems and their connection problem

Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations

We give a unified interpretation of confluences, contiguity relations and Katz’s middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representations and series expansions of their solutions are also within our interpretation. As an application to Fuchsian differential equations on th...

Galois theory of fuchsian q-difference equations

We propose an analytical approach to the Galois theory of singular regular linear q-difference systems. We use Tannaka duality along with Birkhoff’s classification scheme with the connection matrix to define and describe their Galois groups. Then we describe fundamental subgroups that give rise to a Riemann-Hilbert correspondence and to a density theorem of Schlesinger’s type.

Character Theory of Symmetric Groups, Analysis of Long Relators, and Random Walks

We survey a number of powerful recent results concerning diophantine and asymptotic properties of (ordinary) characters of symmetric groups. Apart from their intrinsic interest, these results are motivated by a connection with subgroup growth theory and the theory of random walks. As applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, as well as a finite...

Nonlinear Perturbations of Fuchsian Systems: Corrections and Linearization, Normal Forms

Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a u...