I will describe an ongoing research project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its space of stability conditions. More specifically we expect to find a complex hyperkahler structure on the total space of the tangent bundle. These ideas are closely related to the work of Gaiotto, Moore and Neitzke from a decade ago. The main analytic input is a class of Riemann-Hilbert problems involving maps from the complex plane to an algebraic torus with prescribed discontinuities along a collection of rays.

Dobrushin (1972) showed that, at low enough temperatures, the interface of the 3D Ising model - the random surface separating the plus and minus phases above and below the $xy$-plane - is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height $M_n$ on a box of side length $n$ is $O_P(\log n)$. We study this interface and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to analyze the maximum height $M_n$ of the interface, and prove that at low temperature $M_n/\log n$ converges to $c\beta$ in probability. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.
Joint work with Reza Gheissari.