saiei@educ.kumamoto-u.ac.jp
In this talk, we study a hypergeometric integral associated with any polynomial. When the polynomial is Newton non-degenerate, such an integral is a solution to the Gelfand-Kapranov-Zelevinsky system, which is by now a well-studied holonomic system. On the other hand, most hypergeometric integrals come from degenerate polynomials. Important classes include Feynman integrals in quantum field theory and marginal likelihood integrals in statistics. A twisted cohomology group is a system of difference equations satisfied by hypergeometric integrals. A remarkable feature is that it naturally "converges to" likelihood equations previously studied in algebraic statistics. As an application of this relation, we will see that solving likelihood equations has two interpretations. The first one is that it computes a basis of the twisted cohomology group. The other one is dual to the first one: it computes a basis of Lefschetz thimbles. This is a joint work with Simon Telen (MPI MiS), arXiv:2301.13579.