25 problems

Let $X$ be a homogeneous space, and let $QH^*(X,\mathbb{Z})_{loc}$ denote its quantum cohomology ring localized by inverting the quantum parameter. The homogeneous space $X$ is cal…

Fan's conjecture. The algebra $TL(H_n)$ is generically semisimple.

Let $H$ be a finite-dimensional quasi-Hopf algebra with structure elements $\alpha$ and $\beta$, antipode $S$, and Drinfeld double $D(H)$. Let…

Let $R_{\sigma\tau}^{(n)}(t,\lambda)$ be the coefficient of $\hbar^n$ in the asymptotic solution matrix $R_{\sigma\tau}(t,\lambda,\hbar)$ of the equivariant quantum differential eq…

Assume that the equivariant quantum cohomology has convergent structure constants around $x=\lambda=0$, so that the canonical coordinates $\mathsf{u}_\sigma(t,\lambda)$ are defined…

Let $G$ be the group in the paper, let ${\mathcal I}_G$ be the indicated category of $G$-modules, and let $N$ denote the filtration used to define $gr^N_jW$. Semisimplicity conject…

Assume $p\nmid(e+1)$, let $I=\\{0,1,\ldots,e\\}$, and let $R^{\Lambda}_n(I)$ denote the cyclotomic quiver Hecke algebra associated with a dominant integral weight $\Lambda\in P^+$.…

Let $n\in\mathbb{N}$, let $\mathbb{K}$ be an algebraically closed field, and let $\mathcal{H}^{f}_{\mathbb{K}}$ be the cyclotomic Hecke-Clifford superalgebra associated with…

Let a rational vertex operator algebra (VOA) be given, and consider the category in the preceding semisimplicity statement for integrable factorization modules. Semisimplicity conj…

Let $X$ be an adjoint variety of type $A_n$, and let $Y\subset X$ be a general hyperplane section. Let ${\rm QH}(X)_{\rm can}$ and ${\rm QH}(Y)_{\rm can}$ denote the restrictions o…

Let $X$ be an adjoint or quasi-minuscule variety as above, and let $Y\subset X$ be a general linear section. Write ${\rm BQH}(Y)$ and ${\rm QH}(Y)$ for its small and big quantum co…

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and let $f$ be a polarized self-morphism as in the generalized Weil's Riemann hypothesis:…

Weakened semisimplicity conjecture $({\rm Semi}_{\leq n}')$. For any integer $d$ and morphism $f:M\rightarrow N$ in ${\rm MM}_{\leq n}$ satisfying either

Let $X_0$ be a smooth, separated, geometrically connected scheme over a finitely generated field $k_0$ of characteristic $p\geq 0$, let $f_0:Y_0\to X_0$ be a smooth proper morphism…

Let $\mathfrak{g}$ be a semisimple Lie algebra. Semisimplicity conjecture. The category of finite-dimensional $D(U_q(\mathfrak{g}))$-modules is semisimple. The source attributes th…

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $U_q(\mathfrak{g})$ be its quantized enveloping algebra with quantum double $D(U_q(\mathfrak{g}))$. A finite-dimensional rep…

Beilinson's semisimplicity conjecture. The homology objects of motives of smooth projective varieties over $K$ with respect to $t_l$ are semisimple in the heart…

Let $F$ be the ambient field with absolute Galois group $G$, and for a $G$-module $W$ define the level filtration $N_{\bullet}=N_{\bullet}^{(k)}$ by $N_jW=\sum_{F'}W^{G_{F|F'}}$, w…

Let $V^{\bullet}$ be the sheaf constructed in the paper from closed differential forms and related quotients. Semisimplicity conjecture. The sheaf $V^{\bullet}$ is semisimple. The…

Let $H^{\bullet}=\bigoplus_{q\geq 0}H^q$ be a cohomology theory regarded as a dominant ${\mathbb A}^1$-presheaf, and let $\underline{H}^{\bullet}_c$ be the dominant ${\mathbb A}^1$…

Let $X$ be a rational homogeneous space with Picard number one, and let $QH^*(X,\mathbb{C})_{loc}$ denote its quantum cohomology algebra localized by inverting the quantum paramete…

Let $K$ be a finitely generated field, let $A$ be the coefficient ring, and let $M$ be a semisimple $A$-motive over $K$. For a prime $\mathfrak{p}$ of $A$, write…

Let $\mathfrak{g}$ be a simple Lie algebra, let $f$ be a nilpotent element, and call $(k,f)$ exceptional when the Euler–Poincaré character of the corresponding reduced module satis…

Conjecture PB. The vertex algebra $W_k(\mathfrak{g})$ is semisimple if and only if $k$ satisfies these conditions. This asserts that the displayed exceptional levels are exactly th…

Semisimplicity conjecture. (i) If $n=2k+1$, then $B_n(\hat{z})$ is semisimple only if $\kappa\mathfrak{S}_n$ is semisimple and $p_{2k-1}(\hat{z})\ne 0$. (ii) If $n=2k$, then…