Computing with any sort of object requires a way of encoding it on a computer, which poses a problem in enumerative combinatorics where the objects of interest are (infinite) sequences. Thankfully, the generating function of a combinatorial sequence often satisfies natural algebraic/differential/functional equations, which can then be viewed as data structures for the sequence. The field of analytic combinatorics asks which properties of a sequence are decidable from such encodings and, for those that are, how fast they can be determined.

We begin this lecture series describing how methods from complex analysis, commutative algebra, and validated numerics combine to create effective methods which compute asymptotic behaviour for many types of univariate sequences. We then turn to the newer area of analytic combinatorics in several variables (ACSV), which has been developed to examine multivariate sequences and their multivariate generating functions. Using tools from complex analysis in several variables, topology, and computational algebraic geometry, ACSV not only allows for a study of multivariate behaviour (including limit theorems) but also provides new pathways to attack longstanding decidability questions on the asymptotics of certain families of univariate sequences.

In addition to surveying the relevant theory, we will see current implementations of these methods in computer algebra systems (mainly SageMath) and discuss applications in combinatorics, the analysis of algorithms, and a variety of mathematical and scientific domains.

There is limited funding available for poster presenters and contributed talks. The application deadline is March 31, 2026. Notifications about acceptance of posters and talks are sent out in April. While the topics of the contributed talks should have a direct connection to the topic of the lecture series, the posters may be connected to (analytic) combinatorics in a broad sense.