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 $d_0=0$.

Optimality conjecture. The exponent $\gamma'=\frac{\gamma}{1-\gamma}$ is optimal.

Gross's conjecture. The sharp version of this inequality should have the same constants as those found by Gross in the classical case.

Conjecture. The bound (5.7) could remain valid assuming only the finiteness of $\tau(e^{-\beta h})$. This concerns extending the stated estimate beyond the hypotheses used to deriv…

Let $U_r\in\mathbb{R}^{d\times r}$ be an orthonormal matrix, and let $\mathrm{Gr}(d,r)$ denote the Grassmann manifold of $r$-dimensional subspaces of $\mathbb{R}^d$. Define … as in…

Let $G$ be a Carnot group, let $K$ be a non-smooth homogeneous norm on $G$, and consider probability measures with density $e^{-U}$ with respect to Haar measure, where $U=U(K)$. Su…

Self-adjoint Lindbladian conjecture. For every self-adjoint Lindbladian $L$,

Complete CLSI anti-transference conjecture. The corresponding unregularized constant should satisfy

Model-density conjecture. The model densities on which $\bar \lambda_{LS}[\mathsf{CD}_{1}(K,N),D]$ is attained are the same as the model densities determining the sharp constants…

Dimension-free transportation approximation conjecture. A dimension-free analogue of the cited transportation-structure theorem should hold; combined with the paper's bound, this w…

Laplace-Beltrami CLSI conjecture. On every compact Riemannian manifold, the Laplace-Beltrami operator satisfies $\lambda$-CLSI.

Let $(\mathcal{X},Q,\pi)$ be a Markov chain with invariant measure $\pi$, and suppose that a concentration property with respect to the distance $d_{\mathcal{W}}$ holds with profil…

Let $A_{\theta}^{\infty}$ be the smooth noncommutative two-torus with canonical trace $\tau$, generated by unitaries $U,V$, and let…

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…

Optimal-scaling LSI conjecture. The optimal scaling of the LSI constant holds for $\mu_{N,m}$ under this assumption on $\psi$.

Let $\mathcal{L}:\mathcal{M}_d\rightarrow\mathcal{M}_d$ be a primitive Liouvillian. A Liouvillian is weakly and strongly $\mathbbm{L}_p$-regular when it satisfies, respectively, th…

Let $(M,g)$ be a gradient shrinking Ricci soliton, and let $mu_s$ denote its logarithmic Sobolev invariant. Rigidity conjecture. If … then the shrinker is isometric to…