7 problems

Gap conjecture. For any collection of plumbed $3$-manifolds $Y$, gaps between consecutive $q$-series exponents at the beginning of the series are statistically more significant tha…

Let a plumbed 3-manifold be represented by an arbitrary, possibly non-reduced, plumbing, and let the associated $(q,t)$-series be specialized by taking $t\rightarrow 1$ to obtain a…

Let $\hat{Z}_a(q,t)$ be the refined invariant associated with a plumbed 3-manifold and a $\operatorname{Spin}^c$ structure $a$, and suppose that the recovered series $\hat{Z}_a(q)$…

Gukov's conjecture. The function $\widehat{Z}_{\boldsymbol{a}}(q)$ is a depth $r$ quantum modular form whose quantum set is a subset of $\mathbb Q$. Moreover, for any unimodular pl…

RT–$\hat{Z}$ correspondence. The displayed formula is true for all plumbed 3-manifolds. The formula extends the relation between WRT invariants and $\hat{Z}$ invariants from…

Derived-groups conjecture. The groups $A'$ and $A”$ are isomorphic, up to a grading shift, to the groups $B'$ and $B”$, respectively.

Let $G$ be a plumbing tree, and let $Y_G$ be the corresponding plumbed $3$-manifold. Denote the lattice homology of $G$ by $\mathbb{HF}^{-}(G)$ and the Heegaard Floer homology of…