33 problems

Zagier's conjecture. With respect to the modular transformation $x\mapsto 1/x$, the function $h(x)$ can be extended to a function on $\mathbb{R}$ that is continuous at every irrati…

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

Let $g=2$ and let $x=r/p$ be a rational number with coprime odd integers $1\leq r<p$. Write $\sigma_2(x)\in\mathbb{Z}$ for the signature associated with the corresponding…

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 Hurwitz conjecture. For every real $x>0$, the radius of convergence satisfies

A modular resurgent series is an asymptotic series whose Borel plane has the modular resurgent structure introduced in the paper. Effectiveness conjecture. Median resummation of mo…

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…

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…

Let $M_3$ be a Seifert manifold with three singular fibres. For $b\in{\rm Spin}^c(M_3)$ and defect labels $\boldsymbol\nu\in\mathbb N^3$, define … Identify $q$-series up to multipl…

Let $B$ be a nonsingular plumbing matrix with $|\det B|>1$, let $q=e^{2\pi i/k}$, and let $Y$ be the associated plumbed 3-manifold. Set … where $a\in\mathbb{Z}^V/B\mathbb{Z}^V$ and…

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…

Let $K$ be a knot. Let $J^K:\mathbb Q/\mathbb Z\to\overline{\mathbb Q}$ be a Galois-equivariant function whose value at $-1/n$ is $\langle K\rangle_n$. 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.

Quantum modular depth conjecture. The invariant $\widehat Z^G$ is related to a quantum modular form of depth $nr$.

The homological invariants $\widehat Z$ are invariants associated with three-manifolds, and a quantum modular form is a function on the rationals whose failure to be modular is sui…

Zagier's one-sided-limit conjecture. These left and right limits exist at every rational $r\in(0,1)$; equivalently, the corresponding one-sided limits exist for $\psi$.

The modularity conjecture. For general lens spaces, the homological blocks are conjectured to satisfy

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…

Zagier's modularity conjecture. There exist $C_K(\alpha)\in\mathbb{C}$ and complex numbers $D_{K,n}(\alpha)$ such that, for every $M\in\mathbb{N}$ and rational $x\to\infty$,

Zagier's modularity conjecture. The function $x\mapsto\log J_{4_1,0}({\rm e}(x))$ satisfies a certain reciprocity formula of weight $0$; in particular, the displayed difference is…

Gukov's conjecture. For any tree and labeling such that $M$ is positive definite, $\widehat{Z}_{{\boldsymbol{a}}}(q)$ is a quantum modular form.

Let $K\subset S^3$ be a knot with complement $M=S^3\backslash K$, and define, for a rational number $\alpha=a/c$, $\mathcal{J}_K(\alpha):=J_K(c,e^{2\pi i a/c})$. Let…

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 $k$ be a positive integer, let $\zeta_k=e^{2\pi i/k}$, let $F_K$ be the trace field of $K$, and let $F_{K,k}=F_K(\zeta_k)$. Suppose…