16 problems

Let $(M,g)$ be a compact Riemann surface with volume $1$, let $u_k$ be a sequence of blow-up solutions of the mean field equation with $S=\emptyset$, and let $\mathcal{B}$ be its f…

Let $u$ be a smooth function on the standard sphere $\mathbb{S}^{2}$, and let ${\Greekmath 010B}$ be a positive constant satisfying … A solution is axially symmetric if it is invar…

Lin's axial symmetry conjecture. If $\gamma>0$ and $\beta\leq 16\pi$, then every solution is axially symmetric with respect to $\vec n$.

Conjecture 1. There exists $E_{\Omega}>E_d$ such that the displayed inequality holds for every $E>E_{\Omega}$. This is a proposed high-energy strict-convexity property of the entro…

The state of neuron $j$ is $u^j_t\in\mathbb{R}^c$, and the state of the directed edge from neuron $k$ to neuron $j$ is $z^{jk}_t\in\mathbb{R}^d$. The network has separate inhibitor…

Let $\nu$ be a probability measure on $\mathbb{R}^3$, let $\bar{\nu}$ be its $(0,1)$-marginal, and let $\pi$ and $\bar{\pi}$ denote the invariant measure of the Markov local-field…

Let $\rho_t$ and $\mu_t$ denote, respectively, the laws at time $t \geq 0$ of solutions to the local-field equation (LFE) and the Markov local-field equation (MLFE). Let $d$ be a s…

Heterogeneous-network VFPE conjecture. For heterogeneous networks, the Vlasov–Fokker–Planck equation associated with the stochastic network should be

Evenness and axial symmetry conjecture. For $4<\lambda\leq6$, every solution of the mean field equation must be even, and hence axially symmetric.

Let $E_{\tau}:=\mathbb{C}/\Lambda_{\tau}$ be the flat torus, let $G(z\mid\tau)$ be its Green function, and define the multiple Green function … Count critical points of…

Let $\widetilde{u}(t,z)$ solve the Ornstein–Uhlenbeck Fokker–Planck equation … obtained by replacing the coefficients in the Fokker–Planck equation (FP) with their linear approxima…

Let $\Omega$ be a convex domain of second kind. Let the conclusions of Theorem 3 denote the properties established there for the entropy and mean field equation in the paper. Conje…

Lin–Lucia uniqueness conjecture. The function $u\equiv 0$ is the unique solution of $$ whenever

Let $g$ be the standard metric on $S^2$, with volume form $d\omega$, let $P\in S^2$, and let $\delta(P)$ denote the Dirac mass at $P$. For $\alpha>-1$ and $\lambda>0$, consider … a…

McKean's conjecture. The one-dimensional mean-field equation is the Burgers equation.

Sign-count conjecture. There are $n$ branch points on the associated hyperelliptic curve for which $D(S)<0$ and $n+1$ branch points for which $D(S)>0$.