Abstract: In the realm of multivalued functions, certain specimens run the risk of being elementary or complex to a fault. The Lambert W function serves as a middle ground in a way, being non-representable by elementary functions yet admitting several properties which have allowed for copious research. W utilizes the inverse of the elementary function xe^x , resulting in a multivalued function with non-elementary connections between its branches. W_k(z), the solution to the equation z = W_k(z)e^W_k(z) for a “branch number” k ∈ Z, has both asymptotic and Taylor series for its various branches.

This lecture will primarily focus on the asymptotic series approximations of any branch of the lambert W function. The series previously used to approximate the "principal" branch of the Lambert W can be generalized to any branch. Similarly, the presentation will focus on improvements to these series and why they're possible. Also included is a section on a family of functions closely related to Lambert W, and how their asymptotic approximations can be derived from W's.

Abstract: Counting partitions in diverse dimensions is a long-standing problem in enumerative combinatorics. It also plays a prominent role in the physics of instanton counting and in algebraic geometry through the computation of Donaldson-Thomas invariants. In this talk I will give an overview of these counting problems, and discuss how recent developments in the computation of instanton/Donaldson-Thomas partition functions clarify some open problems in the enumeration of higher-dimensional partitions.