63 problems

Let $Z\subset\mathbb{R}$ be a nonempty finite set, let $g:Z\to\mathbb{R}$, let $Y\subset\mathbb{R}$ be a nonempty finite set, and let $f:Y\to\mathbb{R}$ satisfy … For $z\in Z$, def…

Mosco-convergence conjecture. For every fixed $\bar\rho\in\mathcal P_2^{\mathcal S}(\mathbb R^d)$,

Let $n(r)$ be the number of zeros of the planar Gaussian analytic function $\mathbf g$ in the disk $D(0,r)$. Sodin's deviation conjecture. As $r\to\infty$, … This predicts the loga…

Let $k\geq 2$ be an integer, let $\chi$ be a real non-principal Dirichlet character modulo $q$, and let $f=\mu^{(k)}\chi$ or $f=\mu^{(k)}g_\chi$. Then the large-values limsup conje…

Let $M(x)=\sum_{n\leq x}\mu(n)$ denote the summatory Möbius function. Ng's large-deviation conjecture. There exists a constant $B>0$ such that … This conjecture concerns the precis…

Let $(X,\Delta)$ be a log Fano manifold admitting a Kähler–Einstein metric. Let $Z_{\boldsymbol{m}_{N}}$ be the K-reduced partition function, let $\mathcal{M}$ be the Mabuchi funct…

Conditional law of large numbers conjecture. The convergence in the conditional law of large numbers holds with

2SRWB rate-function ansatz. For each $x\in[0,1]$, there is a regular function $I^{\mathrm{2SRWB}}$ such that

The log-Gamma polymer large deviation conjecture. There exists $\theta_0>0$ such that uniformly for $0<\theta<\theta_0$ and $\epsilon<s<-\theta\psi(\theta)$ for any $\epsilon>0$, w…

Let $\mu\in\mathscr{N}_+$ and write $v:=\partial_x g:=\partial_x\mathsf{H}[\mu]$, where $v$ is a tractable solution. Non-entropy shock comparison conjecture. The measure $\mu$ rest…

Let $D$ be the space of admissible directed-landscape environments, let $I$ be the associated rate function, and let $Q$ be the Dirichlet energy of a function…

The directed-landscape geodesic upper-tail rate conjecture. For $t\in (0,1/2]$, as $a\to\infty$,

Triangle-free graph discontinuity conjecture. The function

Forcing-independence conjecture. The large deviations principle for the system holds independently of the choice of the forcing $f\in L^2(0,T;H_0)$; equivalently, if an LDP holds f…

LDP under the weak Kato hypothesis. The statement of the main large-deviations theorem holds true if the strong Kato hypothesis is replaced with the weak Kato condition.

Theorem 1.1 limit-shape conjecture. The results of Theorem 1.1 hold for all $\vec{\mathbf{r}}\in\mathscr{R}$ under $N\geq1$ and $N^2T\to\infty$.

Let $\mu$ be a probability measure, and let $\rho_\mu:[0,1] \to \mathbb{R}$ denote the right-continuous inverse function of $F_\mu(s):= \int_{-\infty}^s \mu(\mathrm{d}x)$. Let…

Let $\mathcal Z_d^{\text{typ}}$ denote the typical Poisson zero cell, and let $f_0(\mathcal Z_d^{\text{typ}})$ be its number of vertices. Large-deviation conjecture. As…

Let $\{X_{n,i};i=1,\ldots,k_n\}$ be an array of independent random variables in the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with … Let $x_n$ be a se…

Nonlinear Perron–Frobenius conjecture. There exists a value $E^\star(\boldsymbol{\gamma})$ such that: for $E>E^\star(\boldsymbol{\gamma})$, all orbits tend towards the system bound…

Let $X$ be a Fano manifold with volume form $dV$, let $\beta_0<0$, and let $\beta$ range over the real numbers satisfying $\beta>\beta_0$. For each $N$, let $\mathcal{Z}_N(\beta)$…

Let $\tau$ be uniformly distributed on $[T,2T]$, and set … For $k>0$, consider the regime $V\sim k\sqrt{2\log\log T}$. Radziwiłł's conjecture. In this regime, … This predicts that…

Hamiltonian structure conjecture. There exist an energy $\mathcal{E}:\mathcal{Z}\to\mathbb{R}$ and a skew-symmetric operator…

Structural equivalence conjecture. There exist maps $\psi_1$ and $\psi_2$ with the stated diffeomorphism properties such that, for every $(\alpha,\beta,t)\in D$ and $m\in\Delta^2$,

Pathwise asymptotic enumeration conjecture. As $n_N\to\infty$ and $\varepsilon\to0$,