9 problems

Let $Y$ be an integer homology sphere, let $V_q(Y)$ and $V_{\tilde q}(Y)$ be the spaces of $q$-series associated to $Y$, and let $S$ denote the predicted Langlands-duality map on s…

Let $Y$ be a closed $3$-manifold and let the BPS $q$-series obtained by inserting line defects define a homomorphism from the skein module of $Y$ to a space of $q$-series. Holomorp…

Let $M\cong N/\pi_1(M)$ be a geometric rational homology sphere with no $2$-torsion in homology. Let $\xi^{1/4}$ be a primitive $4r$-th root of unity, with…

Let $M\cong N/\pi_1(M)$ be a geometric integer homology sphere with model geometry $N$, and let $W_M(\xi)$ be its properly normalized Witten–Reshetikhin–Turaev invariant for a prim…

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 $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbb Z)$ with $c\neq0$, and let $W_\gamma$ be defined on $\mathbb Q\smallsetminus\{\gamma^{-1}(\infty)\}$.…

Let $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbb Z)$ with $c\neq0$, and let $W_\gamma$ be the cocycle defined on…

Let $r+1$ be the size of the matrix $\mathbf J$, and let $\mathrm{DJ}^{(m)}(q)$ denote the descendant invariant. The matrix $\mathbf J$ is the matrix of the conjectured knot invari…

The t-deformed ADO limit conjecture. The preceding expression should be a polynomial and, at $t=-1$, should equal the usual $p$-th ADO polynomial of $K$ for $SU(2)$. More generally…