12 problems

Jack-polynomial interpolation conjecture. The generating series $\mathcal{R}_\mu(t,\mathbf{p};b)$ interpolates between the quasimodular forms given by the generating functions for…

Quantum modularity conjecture. The $q$-bracket $\langle f\rangle_q$ is a quantum modular form of weight $k$.

Quantum modularity conjecture. This generating function is a quantum modular form.

Let $f\colon\mathbb{Q}\to\mathbb{C}$ be a $q$-series with $q={\rm e}^{2\pi {\rm i}y}$. Suppose its asymptotic expansion $\tilde{f}(y)$ as $y\to0$ extends to $y\in\mathbb{C}$ and ha…

For $n\geq 0$, let $(x;q)_n=(1-x)(1-xq)\cdots(1-xq^{n-1})$ be the $q$-Pochhammer symbol, and let $\begin{bmatrix}n\k\end{bmatrix}_q$ denote the $q$-binomial coefficient. For…

Quantum modular depth conjecture. The invariant $\widehat Z^G_{\vec{\underline{b}}}(M_3;\tau)$ is related to quantum modular forms of depth up to $r\times n$.

General quantum modularity conjecture. The invariant $\widehat Z^G_{\vec{\underline{b}}}(M_3)$ is closely related to a quantum modular form of some kind.

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…

Let $\uptheta\in\mathbb R-\mathbb Q$ be a fundamental quadratic unit and let $K=\mathbb Q(\uptheta)$, with fundamental discriminant $D$. The multivalued quantum modular invariant…

Let $K$ be a knot, let $\zeta$ be a root of unity, and let $\gamma$ be a non-degenerate Neumann–Zagier datum arising from a triangulation of the knot complement. Topological invari…

Andrews–Garvan–Straub conjecture. There exists a $B\in\mathbb N$ such that

For fixed integers $a,b,A,B$, let $f_{a,b,A,B,h,k}$ be the modular forms used to cancel the singularities of $g_2(\zeta_b^a q^A;q^B)$ as $h/k$ ranges over $\mathbb{Q}$. Rhoades' fi…