159 problems

Let an $n$-dimensional quantizer be a quantization scheme in $\mathbb{R}^n$, and distinguish lattice quantizers from arbitrary quantizers. Gersho's conjecture. For all sufficiently…

Let $\hbar$ be the semiclassical parameter, let $\theta_{\mathrm{B}}$ be the Bloch angle, and let $A_{\mathrm{ZJJ}}$ and $B_{\mathrm{ZJJ}}$ denote the functions entering the Zinn-J…

Let ${\rm G}$ be a reductive group with Langlands dual ${rm G}^ atural$, let ${mathbb S}$ be a genus $g$ surface with $n0$ punctures, and let ${cal M}{g,n}$ denote the moduli space…

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…

The paper considers the construction of the operator ${\bf S}$ and the associated quantization theorem for a complex simple Lie algebra ${\mathfrak g}$, with the preceding treatmen…

Let $\mathcal{M}^{G}$ be the moduli space associated with a simple group $G$, and let the star product be the product defined by equation on the chord-diagram algebra. Descent conj…

Let $\mathfrak{g}$ be a simple Lie algebra with $\mathfrak{g}\neq sl(n)$, and let $S\mathfrak{g}$ be the function algebra on the coadjoint representation $\mathfrak{g}^*$. A one-pa…

Let $Y$ be a contact manifold, let $F$ be a vector bundle over $Y$, and let ${\cal J}_1$ and ${\cal J}_2$ be polarizations corresponding to fillings $X_1$ and $X_2$. Assume that…

Consider quantizer designs in which local observations are mapped to quantizer outputs, with stationary or blockwise stationary designs allowed. A quantizer is likelihood-based whe…

In Case A of sequential decentralized detection, each local decision has the form $\phi_n(X_n)$, where $X_n$ is the current observation and no local memory or feedback is available…

Exact-optimality conjecture. For sequences $(\alpha_n)$ of exactly $L^r$-optimal $n$-quantizers, this limiting result holds true, namely

Let $(\alpha_n)_{n\ge 1}$ be an $L^r$-rate-optimal sequence of $n$-quantizers for a probability measure $P$, and consider the lower bound for its $L^s(P)$-quantization rate of conv…

True-rate conjecture. For $m\geq3$, this upper rate is also the true rate of $e_{N,r}(FBS)$. The source notes that the true rate is known for $m=1$ and $m=2$, while the higher-dime…

Let $X$ be a Gaussian process for which the preceding upper bound gives a rate for the product quantization error $e^{(\mathrm{prod})}_{N,r}(X)$ as $N\to\infty$. True-rate conjectu…

Let $B$ be a local commutative algebra over $k$ with maximal ideal ${\mathfrak m} \subset B$, and let $\widetilde{A}$ be a flat associative algebra over $B$ such that…

Conjecture on the relation between $\Psi$ and $J'$. The quantization $\Psi$ of the PL dynamical $r$-matrix $r_0-r'$ is related to the pseudotwist $J'$ by

Let $\mathbb{L}$ be a space equipped with a strong homotopy Lie $1$-algebroid structure $H$, and let $A_1(\mathbb{L})$ denote its associated strong homotopy associative algebra. As…

Existence conjecture. Any modified dynamical $r$-matrix admits a formal twist quantization.

The Hitchin system is an algebraically completely integrable system whose phase space is the cotangent bundle of a moduli space of stable bundles, while the Beauville–Mukai system…

The Hitchin and Beauville–Mukai systems are algebraically completely integrable systems associated with a complex curve $C$ and, in the relevant construction, a spectral transforma…

Let $A$ be a matrix solution of the reflection equation, and let its quantum trace be denoted by $\operatorname{Tr}_q(A)$. A solution is non-degenerate when it is non-degenerate as…

A bisymmetric orbit is an orbit described in the paper's quantization framework by the relevant bisymmetric matrix data. Bisymmetric-orbit quantization conjecture. All bisymmetric…

Let ${\frak h}$ be a Lie algebra, ${\frak g}$ a Lie algebra containing it, and let … be an arbitrary classical triangular dynamical $r$-matrix. Denote by…

Let ${\mathcal O}$ be the nilpotent coadjoint orbit and ${\mathcal M}$ its double cover, with coordinate-ring component $R^{}({\mathcal M})$. For each $x\in{\mathfrak g}$, let…

The paper constructs the algebra ${\mathcal E^{}}^{\lambda_0}$ associated with the double cover ${\mathcal M}$ of a nilpotent coadjoint orbit and considers a corresponding Whittake…