Twisted Cohomology and Likelihood Ideals

In this talk, we study a hypergeometric integral associated with any Laurent polynomial. It is called Feynman integral in physics and called marginal likelihood integral in statistics. A twisted cohomology group is a system of difference equations that hypergeometric integral satisfies. This is a left ideal in a non-commutative ring. It naturally "converges to" likelihood equations previously studied in algebraic statistics. The converse operation exists in principle: the likelihood equation knows the twisted cohomology. We will clarify the meaning of this statement. Based on joint work with Simon Telen (MPI MiS) arXiv:2301.13579.