This talk is about an area known as Analytic Combinatorics in Several Variables (ACSV), which uses complex analytic techniques to obtain asymptotics for combinatorial problems in which a multivariate generating function is known. More abstractly, one solves the problem of estimating the Taylor coefficients of a
multivariable rational function P(z_1, …, z_d) / Q(z_1, … , z_d).

I will begin with a short overview of how it all works, and some examples of the use of ACSV to prove limit shape theorems in combinatorics and statistical mechanics. Next I will discuss the obstacles to automating these analyses, which have to do with explicit computation of cycles in algebraic varieties. Lastly I will present some recent results and conjectures.