Speaker
Description
The main objects of the theory of multidimensional residues are integrals of rational $n$-forms over $n$-dimensional cycles lying in the complement of a polar hypersurface in affine, projective, and toric spaces. Developing ideas of F. Griffiths, V. Batyrev proved (1993) that all periods are $A$-hypergeometric functions in the sense of I. Gelfand, M. Kapranov and A. Zelevinsky (1990) if the differential form is considered by varying all parameters. The Batyrev class can be significantly expanded by moving from integer parameters to complex ones, and instead of compact homology in integration, we can consider cycles with closed (unbounded) supports. Such a generalization can be achieved by considering Mellin transforms in the class of branching integrals (Euler--Mellin integrals). In the last decade, particular interest has arisen in the study of such integrals in connection with the study of Feynman integrals in quantum field theory and string amplitudes in superstring theory. In Bayesian statistics, such integrals appear as marginal likelihood integrals.
The convergence of the Euler--Mellin integral is ensured by the property of quasi-ellipticity of the integrand denominator, first introduced by T. Ermolaeva and A. Tsikh (1996). In the talk we are going to discuss representations of the Euler--Mellin integrals associated with facets of the Newton polytope of the denominator, and their treatments in the context of the theory of Feynman integrals.
