4 problems

Fix a simple Lie algebra $\mathfrak g$ of rank $r$, simple roots $\alpha_1,\dots,\alpha_r$, variables $t=(t_1,\dots,t_r)$, and, for each positive root…

Habiro's conjecture. The loop expansion can be defined near every root of unity $\zeta_m$. This conjecture extends the domain of the loop expansion beyond the formal neighborhood o…

Quantum modularity conjecture. The arithmetically defined power series at each root of unity glue together to elements of the Habiro ring of a number field, allowing one to recover…

Let $K$ be a knot. Let $F_K(x)=\Phi_K(e^{-1/x})$ be the formal expansion of the Habiro element $\Phi_K$ around a root of unity, let $G_K(p)$ denote the Borel transform of $F_K(x)$,…