The asymptotics of the quantum $6j$ symbols corresponding to a hyperbolic tetrahedron is investigated and the first two leading terms are determined for the case that the tetrahedron has an ideal or an ultra-ideal vertex. These terms are given by the volume and the determinant of the Gram matrix of the tetrahedron.

We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings.

On any oriented surface, the affine sphere construction gives a one-to-one correspondence between convex $\mathbb{RP}^2$-structures and holomorphic cubic differentials. Generalizing results of Benoist–Hulin, Loftin and Dumas–Wolf, we show that poles of order less than $3$ of cubic differentials correspond to finite-volume ends of convex $\mathbb{RP}^2$-surfaces, whereas poles of order $3$ and greater than $3$ correspond to geodesic and piecewise geodesic ends, respectively. Moreover, at each pole of order at least $3$, we construct a natural bordification of the surface such that the $\mathbb{RP}^2$-structure extends to the boundary circle in a metric preserving way.

In this paper, we give a generalization of a mathematical definition of Gopakumar–Vafa (GV) invariants on Calabi–Yau 3-folds introduced by Maulik and the author, using an analogue of BPS sheaves introduced by Davison–Meinhardt on the coarse moduli spaces of one dimensional twisted semistable sheaves with arbitrary holomorphic Euler characteristics. We show that our generalized GV invariants are independent of twisted stability conditions, and conjecture that they are also independent of holomorphic Euler characteristics, so that they define the same GV invariants. As an application, we will show the flop transformation formula of GV invariants.