Gauss-Manin connection and higher local fields

The talk is based on a joint project with Victor Przyjalkowski. It will be given a general introdution to Gauss-Manin connection and Picard-Fuchs differential equation defined for algebraic functions on algebraic varieties. Then it will be explained how do higher-dimensional residues of differential forms and higher-dimensional decompositions of rational functions into iterated Laurent series in several variables help to construct solutions of a Picard-Fuchs equation, even in the lack of its explicit description. As a particular case, we will obtain the famous main period formula used in Mirror Symmetry.