40 problems

Let $\mathfrak{h}$ be the Cartan subalgebra and let $\Lambda$ denote the set of parameters for which the corresponding quantization exists, viewed as a semigroup under the semigrou…

Let $(e^{tL})_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $\mathcal{F}$ with generator $(L,D(L))$. Let $G$ be a Chernoff function for $L$ satisfying condition (N) or (N'). Fi…

For $2\leq p<8$, let $t'_p$ denote the optimal time and let $\beta$ be the parameter of the quantum Ornstein–Uhlenbeck semigroup. There are positive constants…

Let $S$ be a finite commutative semigroup with zero, and let $G(S)$ be its zero-divisor graph. A graph is complemented if every vertex is orthogonal to some vertex, where adjacent…

Let $\mathcal{S}$ be a unital commutative semigroup, possibly with the cancellation property, and let $\pi$ be a frame representation of $\mathcal{S}$. Let $\mathcal{A}$ be the…

Let $G_r$ be the nonlinear resolvent associated with $r>0$, and let $q$ and $\gamma_r$ be the parameters defined in the paper. The semigroup generated by $G_r$ is considered with r…

Fix a permutation $w$, and let … Here $\underline\lambda$ and $\underline\mu$ are partition-tuples and $c^w_{\underline\lambda|\underline\mu}$ is the associated coefficient. Semigr…

Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing all archimedean ones. Let $f_1,\ldots,f_s:\mathbb{P}^1\to\mathbb{P}^1$ be rational maps of degree…

Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing all archimedean ones. Let $f_i:\mathbb{P}^1\to\mathbb{P}^1$, $i\in I$, be rational maps of degree…

Freeness conjecture. If $b[?mG_{\mathcal{A}}b[0m$ is a free semigroup and $b[?m\Lambda_{\mathcal{A}}b[0m$ is non-empty and not a singleton, then

Let $\mathfrak G^{(0)}=\{\mathfrak G^{(0)}_\alpha\}_{\alpha\in[\alpha_*,1]}$ be the generalized filtration, and let $f\in\mathfrak G^{(0)}_\alpha$. The function $f$ generates a sem…

Let $G(n,k)$ be a generalized Petersen graph. A semigroup graph is a graph admitting a Cayley representation by a semigroup. The semigroup representation conjecture. Every generali…

Invariant-means conjecture. The set of left invariant means on $S$ coincides with the set of right invariant means on $S$.

Let $\{T_t\}_{t>0}$ be the heat semigroup generated by a Schrödinger operator as above, and let $\{P_t\}_{t>0}$ be the Poisson semigroup subordinated to $\{T_t\}_{t>0}$. Let $X$ be…

Let $\mathsf{T}_{\mathsf{H}}$ be the generator of the $C_{0}$-semigroup $\left(U_{\mathsf{H}}(t)\right)_{t\geqslant0}$, let $\mathbf{P}_{0}$ be the spectral projection associated w…

Remizov's dense second-order approximation-subspace conjecture. The function $G$ has a dense approximation subspace $A^{[0,t_0)}_{\zeta}$ of order $\zeta(n)=1/n^2$, and…

Let $(e^{tL})_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $\mathcal{F}$ with generator $(L,\operatorname{dom}(L))$, and let $G$ be a Chernoff function for $L$. Assume that…

Let $(e^{tL})_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $\mathcal{F}$ with generator $(L,\operatorname{dom}(L))$, and let $G$ be a Chernoff function for $L$. Assume that…

Disconnected-graph conjecture. If $\Gamma_B\backslash\{B\}$ is disconnected, then $B$ belongs to the Ritt ideal.

Fix $q\in\mathbb R$ and let $\mathcal B^D_q$ be the corresponding fiber operator. Set … where $g$ is the parameter in the Bloch–Torrey operator. Let $\lambda_k(q)$ denote the eigen…

Let $X$ be a Banach space, let $d4a5=\left\{F_t\right\}_{t\ge0}\subset C^k(\mathcal{D})$ be a semigroup on a domain $\mathcal{D}\subset X$, and let $T$-continuity mean continuity i…

Let ${\mathbb G}$ be a Carnot group and let $W$ be a horizontal half-space. Denote by $S_W$ the semigroup generated by $W$, by $\bar S_W$ its closure, by $\mathfrak w_{\bar S_W}$ t…

Tripletwist knot-semigroup conjecture. If $(ml+1)n+m$ is odd, then

Vernitski's conjecture. The knot semigroup of every 2-bridge knot is isomorphic to an alternating sum semigroup.

Let $chat K^w_n(t)$ and $chat K^w(t)$ be the random and limiting Robin semigroups from Theorem. For every $t_1,\ldots,t_k>0$, Trace convergence conjecture. … in joint distribution…