13 problems

Let $\sigma^{\ell}$ denote spectral flow by $\ell$, and let $\mathcal{M}$ and $\mathcal{N}$ be $\mathfrak{G}$-modules with fusion product $\mathbin{\times}$. Spectral-flow compatib…

Let $cmathfrak{gl}(1|1)$ be the smallest type-A Lie superalgebra, let $cmathcal V_{chbar,k}(cmathfrak{gl}(1|1))$ be its quantum vertex algebra, and set … Here $csigma^{n,0}$ denote…

Spectral-flow conjecture. For every $\lambda\in\mathrm{Pr}_{\mathbb{Z}}^k$ and $0\leq\theta\leq\bar q$, there exists $\mu\in\mathfrak{h}^*$ such that

Let $\mathsf{k}$ be nondegenerate-admissible. The categories $\mathscr{W}_{\mathsf{u},\mathsf{v}}$ and $\mathscr{W}^{\textup{tw}}_{\mathsf{u},\mathsf{v}}$ consist of weight modules…

In a tight-binding model of graphene, consider a graphene flake with several holes as in the cited work, and adiabatically increase the magnetic flux through the holes. Partial spe…

Booss-Bavnbek–Lesch–Phillips conjecture. The space $\mathcal{FR}^{\mathrm{sa}}(H)$ should be a classifying space for the functor $K^1$.

Consider any system of two linked knots, with two critical points of the zero-mode problem and a line of solutions joining them. Zero-mode line conjecture. For any system of two li…

In the Hopf 2-link, let $p_1,p_2$ be the two critical points in the parameter space, and let the zero-mode problem denote the corresponding equation for zero modes. Initially, the…

Let $V$ and $W$ be admissible level $k$ modules of $\widehat{\mathfrak{sl}}(2)$, let $s,t\in\mathbb Z$, and let $k\in\{-1/2,-4/3\}$. Write $V^s$ for the module obtained from $V$ by…

Twisted Dirac invertible-metric space conjecture. The space $R^{\textup{inv}}_{(F,\nabla)}(M)$ is either empty or has infinitely many path components. This is proposed as a twisted…

Consider the spectral flow of the family of self-adjoint Fredholm Dirac-type operators described above on a two-dimensional domain with $m$ boundary components, where the previousl…

Let $c_m$ denote the coefficient appearing in the spectral-flow formula for the Dirac operator on a compact planar domain with $m$ holes. The conjecture that $c_m=1$. For every pos…

Let $F$ be the Milnor fibre, let $\rho_s\in\pi_0(\operatorname{Symp}(F,\partial F,\omega))$ be its symplectic monodromy, let $D_{P_{\Lambda}(t)}$ be the associated family of operat…