17 problems

Kastoryano–Temme's regularity conjectures. Primitive QMS are weakly $\mathbb{L}_p$-regular with $d_0=0$, and reversible QMS are strongly $\mathbb{L}_p$-regular with $d_0=0$.

Let $\mathbb{Z}_n=\{0,1,\dots,n-1\}$ be the finite cyclic group with normalized counting measure, and let $(P_t)_{t\in\mathbb{R}_+}$ be the Poisson-like semigroup defined by … wher…

Least-atom principle. For all $1<p<q<\infty$,

Critical-extremizer characterization.

Talagrand's convolution conjecture. For every $t>0$ there exists a function $\psi_t:[1,\infty) \to [1,\infty)$ such that $\psi_t(u) \to \infty$ as $u \to \infty$, and for every…

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…

Consider expanding posets, including the Grassmann poset, and simplicial complexes equipped with local-spectral expansion. Special-class conjecture. Simplicial complexes and struct…

Let $X$ be a high-dimensional expander and let $f$ be a function subject to the paper's pseudorandomness condition, except that the condition is weakened by dropping the assumption…

Let the underlying object be a locally spectral expanding structure, such as a simplicial complex or a more general ranked poset, and consider hypercontractivity beyond the second…

Let $E_1$ and $E_2$ be the error parameters for a doubly symmetric binary source, and let $\underline{\Theta}(E_1,E_2)$ and $\overline{\Theta}(E_1,E_2)$ denote the corresponding lo…

Asymptotic optimal-time conjecture. The optimal time to $L_p$-hypercontractivity for $O_N^+$ should be of the form

Let $(X,Y)$ be a doubly symmetric binary source (DSBS), and let $U$ and $V$ be binary random variables generated from $X$ and $Y$, respectively, in the non-interactive simulation s…

Let $n\geq 1$, let $\gamma_n$ be the standard Gaussian probability measure on $\mathbb{R}^n$, and let $P_s$ be the Ornstein–Uhlenbeck semigroup. For every non-negative measurable f…

Let $({\mathcal{X}},T_t,\pi)$ be a Markov semigroup and let its tensorization be denoted by $T_t^{\otimes n}$. Fix $p_0>1$ and $0<\rho_0<(1-p_0^{-1})\ln 2$, and define … Also defin…

Let $n\geq 1$ and $t\geq 0$, let $\mu_t=\frac{1-e^{-t}}{2}\delta_{-1}+\frac{1+e^{-t}}{2}\delta_1$, and let $\mu=\mu_\infty$ be the uniform probability measure on $\{-1,1\}$. Write…

Let $\mu$ be a measure on $\mathbb{R}^n$. A function on $\mathbb{R}^n$ is log-subharmonic if $\ln|f|$ is subharmonic; write $\mathrm{LSH}$ for the class of such functions. The meas…

The configuration space associated with the $\cal B^2_\mu$ $\cal B^2_\mu$ via a $\mu$-deformed Segal-Bargmann transform is $L^2(\mathbf R, |x|^{2\mu}dx)$, with its naturally define…