On Regge–Gel'fand problem of construction of the Pfaff system of Fuchsian type with a given singular divisor

At the beginning of the sixties of the XX century I. M. Gelfand stated the problem of construction of the system of partial differential equations of generalized hypergeometric type for Feynman integrals of quantum electrodynamics. T. Regge presented it on International Conference “Battelle rencontres”. The definition of equations of the generalized hypergeometric type was not given. One of treatment of this statement was the consideration of the problem as the Riemann–Hilbert problem of construction of differential equations using the ramification of a given object and its monodromy group. At this moment (beginning from the twenties) there were known several hypergeometric functions (Appell and Kampe de Feriet) $F_1$, $F_2$, $F_3$, $F_4$ of two complex variables. There was natural to try to write hypergeometric-type equations for them, using the similarity with one-dimensional case. This was done. Next step, using the fact that the Feynman integrals have singular points on the Landau varieties (later the ramification type of these integrals was investigated) and using a theorem of algebraic geometry on reduction of order of pole, the corresponding systems of partial differential equations was written. As in the case of Appell functions of, those equations were of the Fuchsian type. So, the form of desired equations became obvious. From the other hand, physicists prepared for mathematicians very reach class of partial differential equations of Fuchsian type: Knizhnik–Zamolodchikov equations associated with the root system and the ramification along the reflection hyperplanes of these systems. In the structure of the coefficients of such systems the role play the Casimir elements of corresponding Lie algebra. These invariants of the second order and also of higher orders play the principal role in construction of the Fuchsian type equations in two or more parametric case, where as parameters are considered constants number of which is equal to the number of orbits of the root system. It is of great interest the Fuchsian reduction of the nonlinear partial differential equations.
In the talk the results obtained in this direction and open problems will be considered.