50 problems

For a positive integer $n$, let $q$ be the residue-field cardinality and let $s$ be a complex variable. Let $\mathcal{Z}_{n,n,\mathfrak{o}}^{\mathsf{C}}(s)$ denote the symplectic l…

Let $\operatorname{Mod}(S_g)$ be the mapping class group of a closed genus-$g$ surface, and let $\operatorname{Sp}_g(\mathbb{Z}/2)$ be the symplectic group over $\mathbb{Z}/2$. Zim…

Topological irreducibility conjecture. If $\sigma$ is irreducible, then $(\mathscr{N}_\sigma(\Sigma),\pi_\sigma)$ and $(B_{\sigma^*}({\EuScript {P}},V^*),T_{\sigma^*})$ are topolog…

Let $\pi$ be a unitary irreducible representation of $Sp(n,\mathbb R)$ with an integrable infinitesimal character. Let $M_{\pi}(u\otimes v)(g)$ be a $K$-finite matrix coefficient,…

Complete type-edge-pair conjecture. The union over all type-edge pairs equals the solution set:

Type-edge-pair computation conjecture. The folding of $I_1\cup I_2$ gives the entire computed superset $S_1$.

Completeness conjecture. The computed supersets are complete; equivalently, the chambers produced by the relevant type-edge pairs give the entire superset $S_1$.

Let $\mathbf M^{\mathrm{naive}}_{GSp_{2g}}$ denote the naive local model for the symplectic similitude group $GSp_{2g}$ in the ramified setting considered in the paper. Flatness co…

For the compact symplectic group $Sp(n)$, let $\operatorname{cat}(Sp(n))$ denote its Lusternik–Schnirelmann category. Symplectic-group category conjecture. … This is described in t…

Let $R$ be a ring and let $\mathcal{IU}(R^{2n})$ denote the complex of integral unimodular vectors in $R^{2n}$, equipped with the bilinear forms specified in the source. Charney's…

Let $G_n$ and $G_{n-1}$ be the symplectic groups in the preceding branching-rule setting, let $\omega_i^n$ be the relevant fundamental module, and let $D$ be the module occurring i…

Let $\Delta_1$ be the genus-three spin structure specified in the preceding list, and let $Sp(6,\mathbf Z_2)$ act on genus-three spin structures. Consider the generators…

Let $G$ be either the finite symplectic group $\operatorname{Sp}_{N}(q)$ or the finite minus-type orthogonal group $\operatorname{O}^{-}_{N}(q)$, and let $c$ be a Singer cycle in…

Let $F$ be a non-Archimedean local field, let $\boldsymbol{o}$ be its ring of integers, and use the notation $B(\gamma)^{\mathrm{irred}}$, $B(\gamma)^{\mathrm{split}}$, $d_i$,…

Let $N\in\mathbb{N}$, and let $S^*$ denote the unit circle. A function … is $\frac{1}{N}$-admissible if it is a finite sum $\sum_{\vec m}c_{\vec m}(x)z^{\vec m}$, where the exponen…

Shimura's conjecture. The series is a rational function in $X$. This conjecture was proved by Andrianov, so its status is solved.

Universal symplectic determinant invariant conjecture. The canonical map is an isomorphism over every such ring $A$:

Let $\mathbb{F}$ be an infinite field of characteristic not $2$, and for each $n\geq 1$ let $\ell_{4n+2}(\mathbb{F})$ denote the maximum, over elements of the symplectic group in d…

Let $\mathbb{F}$ be a field of characteristic different from $2$, let $V$ be a finite-dimensional vector space over $\mathbb{F}$, and let $b$ be a non-degenerate skewsymmetric bili…

Let $(A,\sigma)$ be a Hermitian algebra, not necessarily semisimple. Write $A^\sigma_{\geq 0}$ for the elements of the fixed-point subspace whose eigenvalues are all non-negative,…

Uniform bounded generation conjecture. There is a minimal constant $K:=K(\Phi)$ proportional to ${\rm rank}(\Phi)$ and independent of $R$ and $I$ such that

Critical symplectic-group conjecture. For these parameters,

Let $X$ be a finite dimensional reduced Stein space, and let $f\colon X\to \operatorname{Sp}_{2m}(\mathbb{C})$ be a holomorphic mapping that is null-homotopic. Let $M_1,\ldots,M_K$…

Let $G$ be the symplectic group in characteristic $2$, let $N$ be the nilpotent element under consideration, let $\mathcal{B}$ be the flag variety of $G$, let $\mathcal{B}_N$ be th…

Let $\mathcal{C}_m(\mathbb{R}^3)$ and the symplectic determinant $D_S:\mathcal{C}_m(\mathbb{R}^3)\to\mathbb{C}$ be defined from the Hopf-lift polynomials $p_\alpha$ using the order…