Lambert series $\sum_{n>0}a(n)q^n/(1-q^n)$ result from a
massive 3-loop banana integral whose algebraic geometry
involves a family of K3 surfaces. Thanks to Ramanujan we
may handle this near the singular limit $q\to1$, where
quasi-modularity leads to a phenomenon called Cheshire cat
resurgence. Danielle Dorogoni and I generalized this to include
resurgent Lambert series with characters, finding applications
to topological string observables [arXiv:2507.21352].
In a study of combinatorics of the cosmohedron
Ardila-Mantilla, et al. found an integer sequence,
1, 2, 10, 72, 644, 6704, 78408, 1008480, 14065744, 210682080...
Its hyper-asymptotic behaviour readily submits to
Jean Ecalle's theory of resurgent functions and alien calculus,
which Michael Borinsky and I used to analyse
infinite series of Feynman integrals [arXiv:2202.01513].