16 problems

Erdős–Simonovits' conjecture. If $e_F\geq e_{F'}$, then, for every graph $G$,

Let $M$ be an $n\times n$ matrix with entries in $\{0,1\}$ and row sums $d_1,\dotsc,d_n$. Minc's conjecture. The permanent of $M$ is at most … Minc's conjecture was proved by Brégm…

Let $(M_t)_{t<T}$ be compact embedded hypersurfaces evolving by mean curvature, with $H>0$, and set $\tau=a-t$ for fixed $a\geq T$. Let $\Omega_t$ be the enclosed domains, let $u$…

Let $F_0$ be a probability density on $\mathbb{R}^d$, and let $F_1$ be a minimizer of the unconstrained variational problem … Here $S$ is the entropy functional and $W_2$ is the…

Let $\lambda$ be the entropy parameter in the stochastic Shigesada–Kawasaki–Teramoto system, currently assumed to satisfy $\lambda>\frac{1}{2}$. Entropy conjecture. We conjecture t…

For the stochastic thin-film equation with mobility exponent $n$ in the range $n\in [8/3,4)$, the existing existence result can be extended to the range $n\in (2,8/3)$ using estima…

Let $g\in H^1(\Omega)^N$ take values in $\mathcal{D}$, prescribe $\rho=g$ on $\Gamma_D\subset\partial\Omega$, and suppose that $s'(g)\in L^\infty(\Omega)$. Define the relative entr…

Let $\mathcal E$ be the set of equilibria of a detailed-balanced reaction system, and suppose that, besides the unique positive detailed-balanced equilibrium $\bm{x}_\infty$, there…

Kleinberg–Sawin–Speyer conjecture. The distribution $\psi_\rho$ belongs to $\Delta_p$. Consequently, the upper bounds $e^{\lambda_p n}$ and $\theta_p^n$ coincide, where $\lambda_p$…

Let $u(t)$ be a solution of the cross-diffusion herding system, with the other parameters fixed as in the preceding exponential-decay theorem. The source excludes $\delta=0$ from t…

The model concerns a volume-surface reaction-diffusion system with stoichiometric coefficients $\alpha$ and $\beta$, and in the degenerate boundary-diffusion case $\delta_v=0$ the…

Let $D$ be a strict graphic sequence of type $\varepsilon$, and let $H_{1}(x)$ be the entropy function whose maximum is attained at $\widetilde{\mathbf{p}}$. Write $\mathbb{G}^{D}$…

Let the entropy production be the dissipation functional associated with the Boltzmann collision operator, and consider a linear inequality relating entropy to entropy production f…

Entropy-based existence conjecture. The existence of such a homogeneous cooling state should follow using only entropy estimates; equivalently, its profile should satisfy

Let a collision kernel have growth order $\gamma \in (-d,+\infty)$ and singularity order $\nu \in [0,2]$, where $\nu=2$ formally plays the role of the Landau collision operator. Le…

Let a collision kernel have growth order $gamma in (-d,+\infty)$ and singularity order $nu in [0,2]$, where $nu=2$ formally plays the role of the Landau collision operator. Villani…