43 problems

Let $M$ be an oriented compact $3$-manifold with a torus boundary, and let $\mathrm{R}^{\rm para}_{\mathrm{SL}_2(\mathbb{C})}(M)$ be the set of conjugacy classes of boundary-parabo…

Let $M$ be an oriented closed $3$-manifold, let $G$ be a simply connected complex simple Lie group, and let $\mathrm{R}^{\rm irr}_G(M)$ be the character variety of irreducible repr…

Asymptotic expansion conjecture. As $N\to\infty$ and $k\to\infty$ with $u$ fixed,

Quantum $SO_3$ spin-Chern-Simons conjecture. The quantum $SO_3$ spin-Chern-Simons theory is, in fact, a TQFT and at the $1$ (mod $2$)-valued $SO_3$ level $2m-1$, it is isomorphic t…

Let $X$ be a closed oriented $3$-manifold and let $G=\operatorname{SU}(2)$. Let $\mathcal M_j$ be the union of the connected components of the moduli space of flat $G$-connections…

For the Deligne–Mumford moduli space of stable curves $\overline{{\mathcal M}}_{g,h}$, let $\Lambda_g^\vee(u)$ denote the total Chern polynomial of the Hodge bundle and let…

Let $\mathfrak{g}$ be a Lie algebra, let $U(\mathfrak{g})[[{\mathchar'26\mkern-9muh}]]$ be its formal power-series enveloping algebra with standard coproduct $\Delta_0$, let $\Phi$…

Expected properties of Yang–Mills plus Chern–Simons theory. The following assertions are expected: (1) the Lagrangian determines a family of Wick-rotated field theories; (2) the si…

Let $\Phi_p^{[j]}(q)^\vee$ be the vector of mock modular forms dual under $q \to 1/q$ to the linear combination of false theta functions $\Phi^{[j]}_p(q)$ associated to the…

Let $\langle W_m\rangle_{\mathcal{N},1,0}^{{\rm SU}(3)_{-4}}$ denote the line defect half-index for the $m$-th Wilson line in the ${\rm SU}(3)_{-4}$ Chern–Simons theory with the in…

Let $K$ be a hyperbolic knot, let $D$ be an open diagram of $K$, and let $\rho$ be a decorated representation of $D$ that is geometrically nondegenerate, meaning that $\rho$ is non…

Let $cmathcal{L}$ be a framed oriented link in $S^3$ with framing $cvec{ctau}=(ctau^1,\cdots,ctau^L)$. For a partition $cmu$, let $cZ_{cmu}(cmathcal{L};q,a)$ be the reformulated fra…

Let $W_2$ denote the two-anyon-pair state, and let $\Sigma_2$ be a genus-$2$ handlebody. The Clifford group orbit of $W_2$ is represented by removing two solid tori from the interi…

Let $M$ be the plumbed $3$-manifold in the preceding asymptotic expansion, let $\mathcal{S}$ be the finite set of phases appearing there, and let…

Let $M$ be a closed and oriented $3$-manifold. For each positive integer $k$, let $Z_k(M)\in\mathbb{C}$ denote the $\operatorname{SU}(2)$ WRT invariant for $M$ at level $k$. Let…

Let $M$ be a cusped hyperbolic $3$-manifold with geometric representation, and let $n$ be odd. Consider the normalized 1-loop invariant…

Let $K$ be a hyperbolic knot. For a parameter $u$, let $\rho_u\colon\pi_1(S^3\setminus K)\to\operatorname{SL}(2;\mathbb{C})$ be the irreducible representation specified by the meri…

Let $L$ be a hyperbolic link, let $J_N(L;q)$ be its colored Jones polynomial, let $\operatorname{Vol}(L)$ be the hyperbolic volume of $S^3\setminus L$, and let…

Let $p\geq 2$, set $\epsilon_1=1$, $\epsilon_2=k+p$, and define the rescaled operators … Let $\mathcal B$ be an appropriate energy eigenbasis for the Hilbert space. Non-Abelian lar…

For an oriented compact three-manifold $X$, let $G$ be a semisimple, simply connected Lie group, and let $\mathcal{Z}_G^{(K)}(X)$ denote the Reshetikhin–Turaev TQFT invariant at le…

Let $H$ be a hyperbolic knot in $S^3$, and let $J_N(H;q)$ be its colored Jones polynomial. Let $\operatorname{CS}(S^3\setminus H)$ denote the $SO(3)$ Chern--Simons invariant associ…

Let $M$ be a connected, oriented, closed Riemannian $n$-manifold with $\HDR^1(M)=0$. Let $\DR(M)$ be its de Rham complex, $\HDR(M)$ its de Rham cohomology, and…

Let $5_2^*$ be the mirror of the knot $5_2$ in Rolfsen’s knot table, and let $r_s$ be the invariant used to study the $\mathit{SU}(2)$-Chern–Simons functional. Irrationality conjec…

Akutsu–Kashaev conjecture. There exists such a family with the following properties:

Let ${\mathfrak g}$ be as in Section 2, let $k\in\mathbb{N}$ satisfy $k>c_{\mathfrak g}$, and let $L$ be a colored simplicial ribbon link in $S^2\times S^1$ as in Theorem 1 or Theo…